Fifteen years ago\, Boe\, Kujawa and the speaker introduced the concept of detecti ng subalgebras for classical Lie superalgebras. These algebras were constr ucted by using ideas from geometric invariant theory. More recently\, D. G rantcharov\, N. Grantcharov\, Wu and the speaker introduced the concept of a BBW parabolic subalgebra. Given a Lie superalgebra \\({\\mathfrak g}\\) \, one has a triangular decomposition \\({\\mathfrak g}={\\mathfrak n}^{-} \\oplus {\\mathfrak f} \\oplus {\\mathfrak n}^{+}\\) with \\({\\mathfrak b }={\\mathfrak f}\\oplus {\\mathfrak n}^{-}\\) where \\({\\mathfrak f}\\) i s a detecting subalgebra and \\({\\mathfrak b}\\) is a BBW parabolic subal gebra. This holds for all classical "simple" Lie superalgebras\, and one c an view \\({\\mathfrak f}\\) as an analog of the maximal torus\, and \\({\ \mathfrak b}\\) like a Borel subalgebra. This setting also provide a usefu l method to define semisimple elements and nilpotent elements\, and to com pute various sheaf cohomology groups \\(R^{\\bullet}\\text{ind}_{B}^{G} (- )\\).

The goal of my talk is to provide a survey of the main ide as of this new theory and to give indications of the interconnections with in the various parts of this topic. I will also indicate how this treatmen t can further unify the study of the representation theory of classical Li e superalgebras.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikhail Zaicev (Lomonosov Moscow State University) DTSTART;VALUE=DATE-TIME:20201022T170000Z DTEND;VALUE=DATE-TIME:20201022T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/16 DESCRIPTION:Title: Polynomial identities: anomalies of codimension growth\nby Mik hail Zaicev (Lomonosov Moscow State University) as part of LieJor Online S eminar: Algebras\, representations\, and applications\n\n\nAbstract\nMikha il Zaicev (Lomonosov Moscow State University\, Russia): Polynomial identi ties: anomalies of codimension growth. Polynomi al identities: anomalies of codimension growth.

Mikhail Zaicev (Lomonosov Moscow State University\, Russia)

22/Oct/2020 - 14:00 GMT-3 (Sã\;o Paulo time)

We consider numerical invariants associat ed with polynomial identities of algebras over a field of characteristic z ero. Given an algebra \\(A\\)\, one can construct a sequence of non-negati ve integers \\({c_n(A)}\, n=1\,2\, \\ldots \\)\, called the codimensions o f \\(A\\)\, which is an important numerical characteristic of identical re lations of \\(A\\). In present talk we discuss asymptotic behavior of codi mension sequence in different classes of algebras.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Natalia Iyudu (University of Edinburgh) DTSTART;VALUE=DATE-TIME:20201119T170000Z DTEND;VALUE=DATE-TIME:20201119T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/17 DESCRIPTION:Title: On the proof of the Kontsevich conjecture on noncommutative birati onal transformations\nby Natalia Iyudu (University of Edinburgh) as pa rt of LieJor Online Seminar: Algebras\, representations\, and applications \n\n\nAbstract\nI will talk about our proof (arxiv 1305.1965\, Duke math J .) of the Kontsevich conjecture (1996) on noncommutative birational transf ormations. It deals with difficulties arising out of the fact that there a re no canonical form for noncommutative rational expressions. Miraculous i dentities proved supposedly reflect some kind of noncommutative group acti ons.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Eli Aljadeff (Technion-Israel Institute of Technology) DTSTART;VALUE=DATE-TIME:20201126T170000Z DTEND;VALUE=DATE-TIME:20201126T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/18 DESCRIPTION:Title: PI theory\, generic objects and group gradings\nby Eli Aljadef f (Technion-Israel Institute of Technology) as part of LieJor Online Semin ar: Algebras\, representations\, and applications\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Agata Smoktunowicz (University of Edinburgh) DTSTART;VALUE=DATE-TIME:20210225T170000Z DTEND;VALUE=DATE-TIME:20210225T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/19 DESCRIPTION:Title: Some questions related to nilpotent rings and braces\nby Agata Smoktunowicz (University of Edinburgh) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nIn around 200 5\, Wolfgang Rump introduced braces\, a generalisation of nilpotent rings to describe all involutive\, non-degenerate set theoretic solutions of the Yang-Baxter equation. This formulation then rapidly found application in other research areas. This talk will review these applications.

Braces h ave been shown to be equivalent to several concepts in group theory such a s groups with bijective 1-cocycles and regular subgroups of the holomorph of abelian groups. In algebraic number theory there is a correspondence be tween braces and Hopf-Galois extensions of abelian type first observed by David Bachiller. There is also connection between R-braces and pre-Lie alg ebras discovered by Wolfgang Rump in 2014. One generator braces have been shown to describe indecomposable\, involutive solutions of the Yang-Baxter equation.

On the other hand\, Anastasia Doikou and Robert West on have recently discovered some fascinating connections between braces an d quantum integrable systems. In particular\, to find solutions of the set -theoretic reflection equation it is needed to solve problems on some poly nomial identities in nilpotent rings. Because previously the theory of pol ynomial identities was mainly developed for prime rings\, and for the refl ection equation we only consider nilpotent rings\, there are no known meth ods for solving such problems. We will mention some open problems on polyn omial identities in nilpotent rings which appear in this situation.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Vladimir Dotsenko (Université de Strasbourg) DTSTART;VALUE=DATE-TIME:20210304T170000Z DTEND;VALUE=DATE-TIME:20210304T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/20 DESCRIPTION:Title: Diamond Lemma and the Maurer-Cartan equation\nby Vladimir Dots enko (Université de Strasbourg) as part of LieJor Online Seminar: Algebra s\, representations\, and applications\n\n\nAbstract\nI shall outline a ne w approach to the Composition-Diamond Lemma for rewriting systems / Gr&oum lbner-Shirshov bases; more specifically\, I shall explain how the Maure r-Cartan equation in the tangent complex of a monomial algebra leads to ma ny different versions of the Composition-Diamond Lemma\, one for each repr esentative of the tangent complex arising from a multigraded resolution of such algebra. This is joint work with Pedro Tamaroff.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexandr Zubkov (UAEU (United Arab Emirates)) DTSTART;VALUE=DATE-TIME:20210311T170000Z DTEND;VALUE=DATE-TIME:20210311T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/21 DESCRIPTION:Title: Harish-Chandra pairs and group superschemes\nby Alexandr Zubko v (UAEU (United Arab Emirates)) as part of LieJor Online Seminar: Algebras \, representations\, and applications\n\n\nAbstract\nThe purpose of my tal k is to discuss the following results recently obtained in collaboration w ith A.Masuoka (Tsukuba University\, Japan). First\, we prove that a certai n category of Harish-Chandra pairs is equivalent to the category of (not n ecessary affine) locally algebraic group superschemes. Using this fundamen tal equivalence we superize the famous Barsotti-Chevalley theorem and prov e that the sheaf quotient of an algebraic group superscheme over its group super-subscheme is again a superscheme of finite type. I will also formul ate some open problems whose solving would bring significant progress in t he supergroup theory.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Maria Gorelik (The Weizmann Institute of Science\, Israel) DTSTART;VALUE=DATE-TIME:20210318T170000Z DTEND;VALUE=DATE-TIME:20210318T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/22 DESCRIPTION:Title: Depths and cores in the light of DS-functors\nby Maria Gorelik (The Weizmann Institute of Science\, Israel) as part of LieJor Online Sem inar: Algebras\, representations\, and applications\n\n\nAbstract\nThe Dul fo-Serganova functors DS are tensor functors relating representations of d ifferent Lie superalgebras. In this talk I will consider the behaviour of various invariants\, such as the defect\, the dual Coxeter number\, the at ypicality and the cores\, under the DS-functor. I will introduce a notion of depth playing the role of defect for algebras and atypicality for modul es. I will mainly concentrate on examples of symmetrizable Kac-Moody and Q -type superalgebras. The talk is based on arXiv:2010.05721\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Apoorva Khare (Indian Institute of Science) DTSTART;VALUE=DATE-TIME:20210325T170000Z DTEND;VALUE=DATE-TIME:20210325T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/23 DESCRIPTION:Title: Polymath14: Groups with norms\nby Apoorva Khare (Indian Instit ute of Science) as part of LieJor Online Seminar: Algebras\, representatio ns\, and applications\n\n\nAbstract\nConsider the following three properti es of a general group \\(G\\):

Algebra: \\(G\\) is abelian and torsion-free.

Analysis: \\(G\\) is a metric space that admits a "nor m"\, namely\, a translation-invariant metric \\(d(.\,.)\\) satisfying: \\( d(1\,g^n) = |n| d(1\,g)\\) for all \\(g \\in G\\) and integers \\(n\\).

While these properties may a priori seem different\, in fact they t urn out to be equivalent (and also to \\(G\\) being isometrically and addi tively embedded in a Banach space\, hence inheriting its norm). The nontri vial implication amounts to saying that there does not exist a non-abelian group with a "norm". We will discuss motivations from analysis\, probabil ity\, and geometry; then the proof of the above equivalences; and fi nally\, the logistics of how the problem was solved\, via a PolyMath proje ct that began on a blog post of Terence Tao.

(Joint - as D.H.J . PolyMath - with Tobias Fritz\, Siddhartha Gadgil\, Pace Nielsen\, Lior S ilberman\, and Terence Tao.)\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Kang Seok-Jin (Korea Research Institute of Arts and Mathematics\, South Korea) DTSTART;VALUE=DATE-TIME:20210401T130000Z DTEND;VALUE=DATE-TIME:20210401T140000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/24 DESCRIPTION:Title: Quantum Borcherds-Bozec algebras and abstract crystals\nby Kan g Seok-Jin (Korea Research Institute of Arts and Mathematics\, South Korea ) as part of LieJor Online Seminar: Algebras\, representations\, and appli cations\n\n\nAbstract\nIn this talk\, we will discuss the basic properties of quantum Borcherds-Bozec algebras and their integrable representations. We also give a brief description of the theory of abstract crystals for q uantum Borcherds-Bozec algebras and their applications.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Evgeny Mukhin (IUPUI School of Science\, USA) DTSTART;VALUE=DATE-TIME:20210408T170000Z DTEND;VALUE=DATE-TIME:20210408T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/25 DESCRIPTION:Title: Supersymmetric analogs of partitions and plane partitions\nby Evgeny Mukhin (IUPUI School of Science\, USA) as part of LieJor Online Sem inar: Algebras\, representations\, and applications\n\n\nAbstract\nWe will explain combinatorics of various partitions arising in the representation theory of quantum toroidal algebras associated to Lie superalgebra gl(m|n ). Apart from being interesting in its own right\, this combinatorics is e xpected to be related to crystal bases\, fixed points of the moduli spaces of BPS states\, equivariant K-theory of moduli spaces of maps\, and other things. This talk is based on a joint project with Luan Bezerra.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/25/ END:VEVENT BEGIN:VEVENT SUMMARY:José María Pérez Izquierdo (Universidad de La Ri oja\, Spain) DTSTART;VALUE=DATE-TIME:20210415T170000Z DTEND;VALUE=DATE-TIME:20210415T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/26 DESCRIPTION:Title: Some aspects of the free nonassociative algebra\nby José María Pérez Izquierdo (Universidad de La Rioja\, Spain) as pa rt of LieJor Online Seminar: Algebras\, representations\, and applications \n\n\nAbstract\nThe free nonassociative algebra provides a simple combinat orial context to extend some constructions from the associative setting. I n this talk\, based on joint work with J. Mostovoy and I. P. Shestakov\, I will briefly discuss three of them related to nonassociative Lie theory: the embedding of the free loop as nonassociative formal power series\, a n onassociative extension of the Baker-Campbell-Hausdorff formula and a nona ssociative version of Solomon's descent algebra.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikhail Belolipetsky (IMPA\, Brazil) DTSTART;VALUE=DATE-TIME:20210422T180000Z DTEND;VALUE=DATE-TIME:20210422T190000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/27 DESCRIPTION:Title: Growth of lattices in semisimple Lie groups\nby Mikhail Beloli petsky (IMPA\, Brazil) as part of LieJor Online Seminar: Algebras\, repres entations\, and applications\n\n\nAbstract\nA discrete subgroup \\(G\\) of a Lie group \\(H\\) is called a lattice if the quotient space \\(H/G\\) h as finite volume. By a classical theorem of Bieberbach we know that the gr oup of isometries of an \\(n\\)-dimensional Euclidean space has only finit ely many different types of lattices. The situation is different for the s emisimple Lie groups \\(H\\). Here the total number of lattices is infinit e and we can study its growth rate with respect to the covolume. This topi c has been a subject of our joint work with A. Lubotzky for a number of ye ars. In the talk I will discuss our work and some other more recent relate d results.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Sergey Shpectorov (University of Birmingham\, UK) DTSTART;VALUE=DATE-TIME:20210429T170000Z DTEND;VALUE=DATE-TIME:20210429T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/28 DESCRIPTION:Title: 2-generated algebras of Monster type\nby Sergey Shpectorov (Un iversity of Birmingham\, UK) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nThe class of non-associa tive axial algebras was introduced in 2015 as a broad generalisation of Ma jorana algebras of Ivanov that were modelled after the properties of the G riess algebra\, the algebra whose automorphism group is the Monster sporad ic simple group. Sakuma's theorem classifies 2-generated Majorana algebras \, which in axial terms correspond to algebras of Monster type (1/4\,1/32) . The quest to classify all 2-generated algebras of arbitrary Monster type \\((\\alpha\,\\beta)\\) was started by Rehren who proved an upper bound o n the dimension and generalised the Norton-Sakuma algebras to arbitrary \\ ((\\alpha\,\\beta)\\). Recently\, new results emerged from the work of Fra nchi\, Mainardis and the speaker\, and independently\, of Yabe\, who class ified symmetric 2-generated algebras of Monster type. Several new classes of algebras have been found.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Natasha Rozhkovskaya (Kansas State University\, USA) DTSTART;VALUE=DATE-TIME:20210506T170000Z DTEND;VALUE=DATE-TIME:20210506T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/29 DESCRIPTION:Title: Generating functions of polynomial tau-functions of the soliton hi erarchies\nby Natasha Rozhkovskaya (Kansas State University\, USA) as part of LieJor Online Seminar: Algebras\, representations\, and applicatio ns\n\n\nAbstract\nThe Kademtsev-Petviashvily (KP) equation is a famous evo lution equation with soliton solutions. It was discovered by M.Sato and th e Kyoto school that the KP equation can be regarded as a part of a countab le system of compatible evolution equations\, which is called today the KP hierarchy. The observation allowed the researchers to discover many new e xamples of soliton type hierarchies and to study them with methods of math ematical physics\, algebraic geometry and representation theory. In the ta lk we will describe the explicit construction of polynomial tau-functions of the KP\, BKP hierarchies through their generating functions. The method uses the tools of representation theory and properties of symmetric funct ions. The talk is based on the joint work with V. G. Kac and J. van de Leu r.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Alistair Savage (University of Ottawa\, Canada) DTSTART;VALUE=DATE-TIME:20210513T170000Z DTEND;VALUE=DATE-TIME:20210513T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/30 DESCRIPTION:Title: Affine Hecke algebras and the elliptic Hall algebra\nby Alista ir Savage (University of Ottawa\, Canada) as part of LieJor Online Seminar : Algebras\, representations\, and applications\n\n\nAbstract\nThe ellipti c Hall algebra has appeared in many different contexts in representation t heory and geometry under different names. We will explain how this algebra is categorified by the quantum Heisenberg category\, which is a diagramma tic category modelled on affine Hecke algebras. This categorification can be used to construct large families of representations for the elliptic Ha ll algebra.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Farkhod Eshmatov (Academy of Science of Uzbekistan\, Uzbekistan) DTSTART;VALUE=DATE-TIME:20210520T170000Z DTEND;VALUE=DATE-TIME:20210520T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/31 DESCRIPTION:Title: On transitive action on quiver varieties\nby Farkhod Eshmatov (Academy of Science of Uzbekistan\, Uzbekistan) as part of LieJor Online S eminar: Algebras\, representations\, and applications\n\n\nAbstract\nThe C alogero-Moser space \\({\\mathcal C}_n\\) is the space of conjugacy classe s of pairs of \\(n \\times n\\) matrices such that the matrix \\(XY - Y X + I_n\\) has rank one. These spaces play important role in geometry\, repr esentation theory and integrable systems. A well-known result of Berest an d Wilson states that the natural action of the affine Cremona group \\(GA_ 2\\) on \\({\\mathcal C}_n\\) is transitive. In this talk we will give a q uiver generalization of this statement and discuss some applications.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Kleshchev (University of Oregon\, USA) DTSTART;VALUE=DATE-TIME:20210527T170000Z DTEND;VALUE=DATE-TIME:20210527T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/32 DESCRIPTION:Title: Irreducible restrictions from symmetric groups to subgroups\nb y Alexander Kleshchev (University of Oregon\, USA) as part of LieJor Onlin e Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nWe motivate\, discuss history of\, and present a solution to the following p roblem: describe pairs \\((G\,V)\\) where \\(V\\) is an irreducible repres entation of the symmetric group \\(S_n\\) of dimension \\(>1\\) and \\(G\\ ) is a subgroup of \\(S_n\\) such that the restriction of \\(V\\) to \\(G\ \) is irreducible. We do the same with the alternating group \\(A_n\\) in place of \\(S_n\\). The latest results on the problem are joint with Pham Huu Tiep and Lucia Morotti.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Vladimir Bavula (The University of Sheffield\, UK) DTSTART;VALUE=DATE-TIME:20210401T170000Z DTEND;VALUE=DATE-TIME:20210401T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/33 DESCRIPTION:Title: The global dimension of the algebras of polynomial integro-differe ntial operators and the Jacobian algebras\nby Vladimir Bavula (The Uni versity of Sheffield\, UK) as part of LieJor Online Seminar: Algebras\, re presentations\, and applications\n\n\nAbstract\nWe review some old and rec ent results about the algebras of polynomial integro-differential operator s and the Jacobian algebras.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Shavkat Ayupov (V.I.Romanovskiy Institute of Mathematics Uzbekista n Academy of Sciences) DTSTART;VALUE=DATE-TIME:20210415T150000Z DTEND;VALUE=DATE-TIME:20210415T160000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/34 DESCRIPTION:Title: Local and 2-local derivations and automorphisms of Octonian algebr as\nby Shavkat Ayupov (V.I.Romanovskiy Institute of Mathematics Uzbeki stan Academy of Sciences) as part of LieJor Online Seminar: Algebras\, rep resentations\, and applications\n\n\nAbstract\nThe talk is devoted to desc ription of local and 2-local derivations (respectively\, automorphisms) on octonian algebras over fields with zero characteristics. We shall give a general form of local derivations on the real octonion algebra \\(O(\\math bb{R})\\). This description implies that the space of all local derivation s on \\(O(\\mathbb{R})\\) when equipped with Lie bracket is isomorphic to the Lie algebra \\(so_7(\\mathbb{R})\\) of all real skew-symmetric \\(7 \\ times 7\\)-matrices. We also consider 2-local derivations on the octonion algebra \\(O(F)\\) over an algebraically closed field \\(F\\) and prove th at every 2-local derivation on \\(O(F)\\) is a derivation. Further\, we ap ply these results to problems for the simple 7-dimensional Malcev algebra. As a corollary we obtain that the real octonion algebra \\(O(\\mathbb{R}) \\) and Malcev algebra \\(M_7(R)\\) are simple non associative algebras wh ich admit pure local derivations\, that is\, local derivations which are n ot derivation. Further\, we shall give a general form of local automorphis ms on the octonion algebra \\(O(F)\\) over a field \\(F\\). This descripti on implies that the group of all local automorphisms on \\(O(F)\\) is isom orphic to the group \\(O_7(F)\\) of all orthogonal \\(7 \\times 7\\)-matri ces over F. We also consider 2-local automorphisms on the octonion algebra \\(O(F)\\) over an algebraically closed field \\(F\\) and prove that ever y 2-local automorphism on \\(O(F)\\) is an automorphism. As a corollary we obtain descriptions of local and 2-local automorphisms of seven dimension al simple Malcev algebra.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Vyacheslav Futorny (IME-USP\, Brazil) DTSTART;VALUE=DATE-TIME:20210603T170000Z DTEND;VALUE=DATE-TIME:20210603T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/35 DESCRIPTION:Title: Infinite-dimensional representations of Lie algebras\nby Vyach eslav Futorny (IME-USP\, Brazil) as part of LieJor Online Seminar: Algebra s\, representations\, and applications\n\n\nAbstract\nWe will discuss the representation theory of simple finite-dimensional Lie algebras\, Affine Lie algebras and their generalizations. Special focus will be given to the representations of vertex algebras.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Pantev (University of Pennsylvania\, USA) DTSTART;VALUE=DATE-TIME:20210610T170000Z DTEND;VALUE=DATE-TIME:20210610T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/36 DESCRIPTION:Title: Geometry and topology of wild character varieties\nby Tony Pan tev (University of Pennsylvania\, USA) as part of LieJor Online Seminar: A lgebras\, representations\, and applications\n\n\nAbstract\nWild character varieties parametrize monodromy representations of flat meromorphic conne ctions on compact Riemann surfaces. They are classical objects with remark able geometric and topological properties. \n\nI will recall how intrinsic geometric structures resolve singularities of wild character varieties an d will show that known algebraic symplectic structures extend naturally to the resolutions. This is based on a new universal method for producing sy mplectic structures which is a joint work with Arinkin and Toen. Time perm itting I may also describe recent joint works with Chuang\, Diaconescu\, D onagi\, and Nawata which extract cohomological invariants of wild characte r varieties from enumerative Calabi-Yau geometry and refined Chern-Simons invariants of torus knots.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Geordie Williamson (University of Sydney\, Australia) DTSTART;VALUE=DATE-TIME:20210617T200000Z DTEND;VALUE=DATE-TIME:20210617T210000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/37 DESCRIPTION:Title: Spectra in representation theory\nby Geordie Williamson (Unive rsity of Sydney\, Australia) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nIn geometric representat ion theory cohomology\, intersection cohomology and constructible sheaves show up everywhere. This might seem strange to an algebraic topologist\, w ho might ask: why this emphasis on cohomology\, when there are so many oth er interesting cohomology theories (like K-theory\, elliptic cohomology\, complex cobordism\, ...) out there? They might also ask: is there somethin g like "intersection K-theory"\, or "intersection complex cobordism"? This is something I've often wondered about. I will describe work in progress with Ben Elias\, where we use Soergel bimodules to investigate what KU-mod ules look like on the affine Grassmannian. We have checked by hand that in types A1\, A2 and B2\, one gets something roughly resembling the quantum group. Speaking very roughly\, the intersection K-theory of Schubert varie ties in the affine Grassmannian should recover the irreducible representat ions of the quantum group. Inspirations for this work include a strange Ca rtan matrix discovered by Ben Elias\, and work of Cautis-Kamnitzer.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Vitaly A. Roman'kov (Sobolev Institute of Mathematics RAS\, Omsk B ranch\, Omsk\, Russia) DTSTART;VALUE=DATE-TIME:20210624T170000Z DTEND;VALUE=DATE-TIME:20210624T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/38 DESCRIPTION:Title: Embedding theorems for solvable groups\nby Vitaly A. Roman'kov (Sobolev Institute of Mathematics RAS\, Omsk Branch\, Omsk\, Russia) as p art of LieJor Online Seminar: Algebras\, representations\, and application s\n\n\nAbstract\nIn this talk\, we present a series of results on group em beddings in groups with a small number of generators. We show that each fi nitely generated group \\(G\\) lying in a variety M can be embedded in a 4 -generated group \\(H\\) in a variety MA\, where a A means the variety of abelian groups. If \\(G\\) is a finite group\, then \\(H\\) can also be f ound as a finite group. It follows\, that any finitely generated (finite) solvable group \\(G\\) of the derived length \\(l\\) can be embedded in a 4-generated (finite) solvable group \\(H\\) of length \\(l+1\\). Thus\, we answer the question of V. H. Mikaelian and A.Yu. Olshanskii. It is also s hown that any countable group \\(G\\) in M\, such that the abelianization \\(G_{ab}\\) is a free abelian group\, is embeddable in a 2-generated grou p \\(H\\) in MA.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Dmitry V. Artamonov (Lomonosov State University\, Moscow) DTSTART;VALUE=DATE-TIME:20210701T170000Z DTEND;VALUE=DATE-TIME:20210701T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/39 DESCRIPTION:Title: \\(3j\\)-symbols for the algebra \\(gl_3\\)\nby Dmitry V. Arta monov (Lomonosov State University\, Moscow) as part of LieJor Online Semin ar: Algebras\, representations\, and applications\n\n\nAbstract\nThe probl em of caculation of Clebsh-Gordan coefficients for a tensor product of two irreducible representations of the Lie algebra \\(gl_2\\) is well-investi gated. It's solution plays an importan role in quantum mechanics. Analogou s problem for the algebra \\(gl_3\\) is also improtant (in the theory of q uarks)\, but it it much l more difficult. In some sence it was solved in t he 60-s in a series of papers by Biedenharn\, Louck\, Baird. But their solution is very cumbersome and not explicit. Thus the problem of findind of an explicit and simple formula for a Clebsh-Gordan coefficient remained unsolved.

In the talk an explicit and simple formula for a Cleb sh-Gordan coefficient for the algebra \\(gl_3\\) will be presented. The a nswer will be given as a value at \\(1\\) of some \\(A\\)-hypergeometric f unction.

As a byproduct I shall give an explicit description of i nvariants in triple tensor product and a projection on the corresponding trivial representation.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikhail Kotchetov (Memorial University of Newfoundland\, Canada) DTSTART;VALUE=DATE-TIME:20210708T170000Z DTEND;VALUE=DATE-TIME:20210708T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/40 DESCRIPTION:Title: Fine gradings on classical simple Lie algebras\nby Mikhail Kot chetov (Memorial University of Newfoundland\, Canada) as part of LieJor On line Seminar: Algebras\, representations\, and applications\n\n\nAbstract\ nGradings by abelian groups have played an important role in the theory of Lie algebras since its beginning: the best known example is the root spac e decomposition of a semisimple complex Lie algebra\, which is a grading b y a free abelian group (the root lattice). Involutive automorphisms or\, e quivalently\, gradings by the cyclic group of order 2\, appear in the clas sification of real forms of these Lie algebras. Gradings by all cyclic gro ups were classified by V. Kac in the late 1960s and applied to the study o f symmetric spaces and affine Kac-Moody Lie algebras.\n\nIn the past two d ecades there has been considerable interest in classifying gradings by arb itrary groups on algebras of different varieties including associative\, L ie and Jordan. Of particular importance are the so-called fine gradings (t hat is\, those that do not admit a proper refinement)\, because any gradin g on a finite-dimensional algebra can be obtained from them via a group ho momorphism\, although not in a unique way. If the ground field is algebrai cally closed and of characteristic 0\, then the classification of fine abe lian group gradings on an algebra (up to equivalence) is the same as the c lassification of maximal quasitori in the algebraic group of automorphisms (up to conjugation). Such a classification is now known for all finite-di mensional simple complex Lie algebras.\n\nIn this talk I will review the a bove mentioned classification and present a recent joint work with A. Eldu que and A. Rodrigo-Escudero in which we classify fine gradings on classica l simple real Lie algebras.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Lubotzky (Hebrew University\, Jerusalem\, Israel) DTSTART;VALUE=DATE-TIME:20210715T170000Z DTEND;VALUE=DATE-TIME:20210715T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/41 DESCRIPTION:Title: First order rigidity of high-rank arithmetic groups\nby Alex L ubotzky (Hebrew University\, Jerusalem\, Israel) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Yuri Berest (Cornell University\, USA) DTSTART;VALUE=DATE-TIME:20210722T170000Z DTEND;VALUE=DATE-TIME:20210722T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/42 DESCRIPTION:Title: Spaces of quasi-invariants and homotopy Lie groups\nby Yuri Be rest (Cornell University\, USA) as part of LieJor Online Seminar: Algebras \, representations\, and applications\n\n\nAbstract\nQuasi-invariants are natural algebraic generalizations of classical invariant polynomials of fi nite reflection groups. They first appeared in mathematical physics --- in the work of O. Chalykh and A. Veselov on quantum integrable systems --- i n the early 1990s\, and since then have found many interesting application s in other areas: most notably\, representation theory\, algebraic geometr y and combinatorics.\n\nIn this talk\, I will explain how the algebras of quasi-invariants arise in topology: as cohomology rings of certain spaces naturally attached to compact connected Lie groups. Our main result is a g eneralization of a well-known theorem of A. Borel that realizes the algebr a of classical invariant polynomials of a Weyl group W(G) as the cohomolog y ring of the classifying space BG of the corresponding Lie group G. Perha ps most interesting here is the fact that our construction of spaces of qu asi-invariants is purely homotopy-theoretic. It can therefore be extended to some non-Coxeter (p-adic pseudo-reflection) groups\, in which case the compact Lie groups are replaced by the so-called p-compact groups (a.k.a. homotopy Lie groups).\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Onofrio Mario Di Vincenzo (Università di Basilicata\, Potenza\, I taly) DTSTART;VALUE=DATE-TIME:20210729T170000Z DTEND;VALUE=DATE-TIME:20210729T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/43 DESCRIPTION:Title: Algebras and superalgebras with (super-)involutions and their poly nomial identities\nby Onofrio Mario Di Vincenzo (Università di Basili cata\, Potenza\, Italy) as part of LieJor Online Seminar: Algebras\, repre sentations\, and applications\n\n\nAbstract\nIn this talk we consider the *-polynomial identities of algebras with involutions. The positive solutio n of Specth's problem\, given by Aljadeff\, Giambruno and Karasik in [E. A ljadeff\, A. Giambruno\, Y. Karasik Polynomial identities with involution\ , super-involutions and the Grassmann envelope\, Proc. Amer. Math. Soc. 14 5 (2017)\, no. 5\,1843-1857]\, for the T*-ideals of the free algebra with involution\, show the decisive role of the identities of finite dimensiona l superalgebras with superinvolution. In this talk we consider block-trian gular matrix algebras related to any sequence of such *-simple superalgebr as. These *-simple superalgebras are also involved in determining the exac t value of the correponding exponent as proved in [A. Ioppolo The exponent for superalgebras with superinvolution\, Linear Algebra and its Applicati ons Amer. Math. Soc. 555 (2018)\, 1-20]. We review the results in this are a and we show that that every minimal affine variety of superalgebras with superinvolution is generated by one of the block triangular matrix algebr as we introduced\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexei Kanel-Belov (Bar Ilan University\, Israel) DTSTART;VALUE=DATE-TIME:20210805T170000Z DTEND;VALUE=DATE-TIME:20210805T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/44 DESCRIPTION:Title: Evaluations of nonassociative polynomials on finite dimensional al gebras\nby Alexei Kanel-Belov (Bar Ilan University\, Israel) as part o f LieJor Online Seminar: Algebras\, representations\, and applications\n\n \nAbstract\nLet \\(p\\) be a polynomial in several non-commuting variable s with coefficients in an algebraically closed field \\(K\\) of arbitrary characteristic. It has been conjectured that for any \\(n\\)\, for \\(p\\) multilinear\, the image of \\(p\\) evaluated on the set \\(M_n(K)\\) of \ \(n\\) by \\(n\\) matrices is either zero\, or the set of scalar matrices\ , or the set \\(sl_n(K)\\) of matrices of trace 0\, or all of \\(M_n(K)\\) .

In this talk we will discuss the generalization of this result for non-associative algebras such as Cayley-Dickson algebra (i.e. algebra of octonions)\, pure (scalar free) octonion Malcev algebra and basic low rank Jordan algebras.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Nikolay Romanovskiy (Novosibirsk State University\, Russia) DTSTART;VALUE=DATE-TIME:20210812T150000Z DTEND;VALUE=DATE-TIME:20210812T160000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/45 DESCRIPTION:Title: Rigid solvable groups. Algebraic geometry and model theory\nby Nikolay Romanovskiy (Novosibirsk State University\, Russia) as part of Li eJor Online Seminar: Algebras\, representations\, and applications\n\n\nAb stract\nA solvable group \\(G\\) is called rigid\, more precisely \\(m\\)- rigid\, if there exists a normal series of subgroups \\(G=G_1 > G_2 > \\ld ots > G_m > G_{m+1}=1\,\\) where all quotients \\(G_i/G_{i+1}\\) are abeli an and when viewed as right modules over \\(\\mathbb{Z} [G/G_i]\\)\, do no t have torsion. Free solvable groups and iterated wreath products of torsi on free abelian groups are rigid\, as well as their subgroups. A rigid gro up \\(G\\) is termed divisible if elements of the quotient \\(G_i/G_{i+1}\ \) are divisible by non-zero elements of the ring \\(\\mathbb{Z} [G/G_i]\\ )\, i.e. \\(G_i/G_{i+1}\\) is a vector space over the skew-field of fracti ons \\(Q(G/G_i)\\) of the ring \\(\\mathbb{Z} [G/G_i]\\) (such a skew-fiel d exists).

The talk will present the results of the author and A. Myasnikov. Among them\, on the algebraic geometry of rigid groups\, we st ate the main two: it is proved that any rigid group is equationally Noethe rian\, and the coordinate groups of irreducible algebraic sets over a divi sible rigid group are described. The theory of models of divisible m-rigid groups is in many ways similar to the classical theory of models of algeb raically closed fields. The axiomatics of the theory of divisible m-rigid groups is found\, \\(\\omega\\)-stability is proved\, saturated models are described\, the elimination of quantifiers is found\, the problems of cal culating the Morley rank are studied. Model theory results use algebraic g eometry over divisible rigid groups.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Eugeny Plotkin (Bar-Ilan University\, Israel) DTSTART;VALUE=DATE-TIME:20210819T170000Z DTEND;VALUE=DATE-TIME:20210819T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/46 DESCRIPTION:Title: Bounded generation and logical properties for linear and Kac-Moody cases\nby Eugeny Plotkin (Bar-Ilan University\, Israel) as part of Li eJor Online Seminar: Algebras\, representations\, and applications\n\n\nAb stract\nWe will survey a series of recent developments in the area of boun ded generation and first-order descriptions of groups. The goal is to illu minate the known results relevant to logical characterizations of Chevalle y and Kac-Moody groups. If time permits I will discuss related questions o riginated from universal algebraic geometry.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Askar Dzhumadil'daev (Academy of Sciences of Kazakhstan\, Kazakhst an) DTSTART;VALUE=DATE-TIME:20210826T170000Z DTEND;VALUE=DATE-TIME:20210826T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/47 DESCRIPTION:Title: Dimension formula for Koszul operads\nby Askar Dzhumadil'daev (Academy of Sciences of Kazakhstan\, Kazakhstan) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nWe g ive recurrence formula for dimensions of Koszul operads. For example\, dim ensions of multi-linear parts of Lie-admissible operad satisfy the followi ng recurrence relations \\(d_n=\\sum_{i=1}^{n-1}\\mu k B_{n-1\,k}(d_1\,\\l dots\,d_{n-1})\,\\) where \\(B_{n\,k}\\) are Bell polynomial and \\(\\mu_k =k!\\sum_{i=0}^k (k-i+1)^i/i!\\). If \\(p>3\\) is prime\, then \\(d_{p-1}\ \equiv 1 (mod p)\,\\) \\(d_{p}\\equiv -1(mod p)\,\\) \\(d_{p+1}\\equiv -1( mod p)\,\\) \\(d_{p+2}\\equiv -6(mod p)\,\\) \\(d_{p+3}\\equiv -56 (mod p) \,\\) \\(d_{p+4}\\equiv -725(mod p).\\)\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Dimitry Leites (New York University Abu Dhabi\, United Arab Emirat es and Stockholm University\, Sweden) DTSTART;VALUE=DATE-TIME:20210902T170000Z DTEND;VALUE=DATE-TIME:20210902T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/48 DESCRIPTION:Title: Classifications of simple Lie (super)algebras and algebras "more i nteresting" than simple\nby Dimitry Leites (New York University Abu Dh abi\, United Arab Emirates and Stockholm University\, Sweden) as part of L ieJor Online Seminar: Algebras\, representations\, and applications\n\n\nA bstract\nI intend to overview classifications of simple Lie (super)algebra s of finite dimension and of polynomial growth. Various properties of comp lex Lie superalgebras resemble same of modular Lie algebras. I will encour age to consider these classifications without fanaticism: certain non-simp le Lie (super)algebras\, "close" to simple ones\, are often "better" for u s than simple ones.\n\nInteresting features of deformations: semi-trivial deformations and (in super setting) odd parameters.\n\nI'll formulate clas sification of finite-dimensional simple complex Lie superalgebras\, odd pa rameters including.\n\nI'll formulate a definition of Lie superalgebra sui table for any characteristic and classification of simple (finite-dimensio nal) Lie superalgebras over algebraically closed fields of characteristic 2. With a catch: modulo (a) classification of simple (finite-dimensional) Lie superalgebras (over the same field) and (b) classification of their gr adings modulo 2. I'll mention conjectures on classification of modular Lie algebras and superalgebras.\n\nIs it feasible to classify simple filtered Lie (super)algebras of polynomial growth? Interesting examples: deforms o f the Poisson Lie (super)algebras\, Lie (super)algebras of "matrices of co mplex size"\, etc.\n\nExamples. Double extensions of simple Lie (super)alg ebras are definitely "more interesting" than the simple objects they exten d.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Maria Ofelia Ronco (Universidad de Talca\, Chile) DTSTART;VALUE=DATE-TIME:20210909T170000Z DTEND;VALUE=DATE-TIME:20210909T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/49 DESCRIPTION:Title: Generalization of dendriform algebras\nby Maria Ofelia Ronco ( Universidad de Talca\, Chile) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nIn a joint work with D. López N. and L.-F. Préville-Ratelle [D. Lopez\, L.-F. Pr&eacu teville-Ratelle\, M. Ronco\, Algebraic structures defined on \\(m\\)-Dyck paths\, preprint arxiv:1508.01252 (2015)] we introduce a family of non-sy mmetric operads \\({\\mbox{Dyck}^m}\\)\, which satisfies that:

1. \\({\\mbox{Dyck}^0}\\) is the operad of associative algebras\,

2. \\({\\mbox{Dyck}^1}\\) is the operad \\({\\mbox{Dend}}\\) of dendriform a lgebras\, introduced by J.-L. Loday in [J.-L. Loday\, Dialgebras\, in Dial gebras and related operads\, Lecture Notes in Math.\, 1763\, Springer\, Be rlin (2001) 7-66]\,

3. the vector space spanned by the set of \\(m \\)-Dyck paths has a natural structure of free \\({\\mbox{Dyck}^m}\\) alge bra over one element\,

4. for any \\(k\\geq 1\\)\, there exist de generacy operators \\(s_i: {\\mbox{Dyck}^m}\\longrightarrow {\\mbox{Dyck}^ {m-1}}\\) and face operators \\(d_j: {\\mbox{Dyck}^m}\\longrightarrow {\\ mbox{Dyck}^{m+1}}\\)\, which defines a simplicial complex in the category of non-symmetric operads.

The main examples of \\({\\mbox{Dyck}^m} \\) algebra are the vector spaces spanned by the \\(m\\)-simplices of cert ain combinatorial Hopf algebras\, like the Malvenuto-Reutenauer algebras a nd the algebra of packed words.

A well-known result on associativ e algebras states that\, as \\({\\mathcal S}\\)-module\, the operad of \\( {\\mbox{Ass}}\\) of associative algebras is the composition \\({\\mbox{As s}} ={\\mbox{Com}}\\circ {\\mbox{Lie}}\\)\, where \\({\\mbox{Com}}\\) is t he operad of commutative algebras and \\({\\mbox{Lie}}\\) is the operad of Lie algebras. The version of this result for dendriform algebras (see [M. Ronco\, Eulerian idempotents and Milnor-Moore theorem for certain non-coc ommutative Hopf algebras\, J. of Algebra 254 (2002) 152-172.])\, is that \ \({\\mbox{Dend}} = {\\mbox{Ass}}\\circ {\\mbox{Brace}}\\)\, where \\({\\mb ox{Brace}}\\) is the operad of brace algebras\, defined in [M. Gerstenhabe r\, A. Voronov\, Homotopy G-algebras and moduli space operad\, Internat. Math. Research Notices (1995)\, 141-153.] and [E. Getzler\, Cartan homotop y formulas and the Gauss-Manin connection in cyclic homology\, Israel Math . Conf. Proc. 7 (1993)\, 65-78.].

Our goal is to introduce the no tion of \\(m\\)-brace algebra\, for \\(m\\geq 2\\)\, and prove that there exists a Poincaré-Birkoff-Witt Theorem in this context\, stating tha t \\({\\mbox{Dyck}^m} = {\\mbox{Ass}}\\circ {\\mbox{m-Brace}}\\).

Joint work with: Muriel Livernet\,Dept. of Mathématiques\, Univ. de Paris-Diderot\, France.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Petr Vojtechovsky (Denver University\, USA) DTSTART;VALUE=DATE-TIME:20210916T170000Z DTEND;VALUE=DATE-TIME:20210916T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/50 DESCRIPTION:Title: Quandles and other classes of set-theoretic solutions of the Yang- Baxter equation\nby Petr Vojtechovsky (Denver University\, USA) as par t of LieJor Online Seminar: Algebras\, representations\, and applications\ n\n\nAbstract\nQuandles are algebraic structures designed to mesh with the Reidemeister moves of knot theory. Joyce and Matveev showed that quandles give rise to a complete invariant of oriented knots. Since the Yang-Baxte r equation resembles the third Reidemeister move\, it is not surprising th at quandles also form a class of set-theoretic solutions of the Yang-Baxte r equation. In this talk I will explain how quandles and connected quandle s can be enumerated up to isomorphism and list a few open problems. I will also present two additional classes (involutive and idempotent) of set-th eoretic solutions of the Yang-Baxter equation with rich algebraic theory.\ n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/50/ END:VEVENT BEGIN:VEVENT SUMMARY:Valery Bardakov (Sobolev Institute of Mathematics\, Novosibirsk\, Russia) DTSTART;VALUE=DATE-TIME:20210923T170000Z DTEND;VALUE=DATE-TIME:20210923T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/51 DESCRIPTION:Title: Quandles and quandle rings\nby Valery Bardakov (Sobolev Instit ute of Mathematics\, Novosibirsk\, Russia) as part of LieJor Online Semina r: Algebras\, representations\, and applications\n\n\nAbstract\nAt the fir st part of my talk I give a definition and examples of racks and quandles\ , explain their connection with knot theory and with set-theoretic solutio ns of the Yang-Baxter equation. Further I introduce some properties of qua ndles: residually finiteness\, orderability\, and formulate results on qua ndles which have these properties.\n\nThe second part of my talk is dedica ted to quandle rings. I introduce generalized quandle ring\, augmented ide al\, describe relationships between subquandles of the given quandle and i deals of the associated quandle ring. The construction of the quotient qua ndle leads to a correspondence between subquandles of the given quandle an d ideals of the quandle ring.\n\nI formulate some results on zero-divisors in quandle rings. Some of these results answer a question of M. Elhamdadi \, N. Fernando and B. Tsvelikhovskiy [J. Algebra\, 526 (2019)\, 166-187] o n quandle rings which do not have zero-divisors.\n\nWe discuss a problem o f the computation of idempotents in quandle rings. The computation of idem potents is then used to determine automorphism groups of some quandle ring s.\n\nI introduce the commutator width of quandle rings and compute the pr ecise commutator width for some quandle rings.\n\nWe also discuss relation s of quandle algebras with other well-known non-associative algebras like alternative algebras\, Jordan algebras and Lie algebras.\n\nAt the end of the talk I formulate some open problems on quandle rings.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Peter Semrl (University of Ljubljana\, Slovenia) DTSTART;VALUE=DATE-TIME:20210930T170000Z DTEND;VALUE=DATE-TIME:20210930T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/52 DESCRIPTION:Title: On Wigner's theorem\nby Peter Semrl (University of Ljubljana\, Slovenia) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nSome recent improvements of Wigner's unita ry-antiunitary theorem will be presented. A connection with Gleason's theo rem will be explained.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael J. Larsen (Indiana University\, USA) DTSTART;VALUE=DATE-TIME:20211007T170000Z DTEND;VALUE=DATE-TIME:20211007T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/53 DESCRIPTION:Title: Quotients of normal subsets in simple groups\nby Michael J. La rsen (Indiana University\, USA) as part of LieJor Online Seminar: Algebras \, representations\, and applications\n\n\nAbstract\nLet \\(G\\) be a fini te simple group and \\(S\\) a normal subset of \\(G\\). If \\(|G|\\) is l arge enough in terms of \\(|S|/|G|\\)\, can we deduce that every element o f \\(G\\) can be expressed as \\(x y^{-1}\\) for \\(x\\) and \\(y\\) eleme nts of \\(S\\)? Shalev\, Tiep\, and I have proven that this is true assum ing \\(G\\) is an alternating group or a group of Lie type in bounded rank \, but the question remains open for classical groups of high rank over sm all fields. I will say something about the methods of proof\, which invol ve both character methods and geometric ideas and also say something about the more general question of covering \\(G\\) by \\(ST\\) where \\(S\\) a nd \\(T\\) are large normal subsets.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Louis Rowen (Bar-Ilan University\, Israel) DTSTART;VALUE=DATE-TIME:20211014T170000Z DTEND;VALUE=DATE-TIME:20211014T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/54 DESCRIPTION:Title: Finitely generated axial algebras\nby Louis Rowen (Bar-Ilan Un iversity\, Israel) as part of LieJor Online Seminar: Algebras\, representa tions\, and applications\n\n\nAbstract\nThis lecture is a continuation of the general talk given at the Drensky conference last month\, on axial alg ebras\, which are (not necessarily commutative\, not necessarily associati ve) algebras generated by semisimple idempotents. After a review of the de finitions\, we investigate the key question\, being\, "Under what conditio ns must an axial algebra be finite dimensional?" Krupnik showed that 3 ide mpotents can generate arbitrarily large dimensional associative algebras ( and thus infinite dimensional algebras via an ultraproduct argument)\, so some restriction is needed. We consider "primitive" axes\, in which the le ft and right eigenspaces having eigenvalue 1 are one-dimensional.

Hall\, Rehren\, Shpectorov solves obtained a positive answer for commutat ive axial algebras of "Jordan type" \\(\\lambda \\neq \\frac{1}{2}\\)\, al though the proof relies on the classification of simple groups and the giv en bound of the dimension is rather high. Gorshkov and Staroletov provided a sharp bound for 3-generated commutative axial algebras of "Jordan type" . Our objective in this project is give a noncommutative version and indic ate how to investigate 4-generated commutative axial algebras of "Jordan t ype"\, in terms of the regular representation.

Our method is to b uild an associative algebra from the adjoint algebra of \\(A\\)\, which ha s a strictly larger dimension which nevertheless also is finite dimensiona l.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Oksana Bezuschak (Kyiv Taras Shevchenko University\, Ukraine) DTSTART;VALUE=DATE-TIME:20211021T170000Z DTEND;VALUE=DATE-TIME:20211021T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/55 DESCRIPTION:Title: Locally matrix algebras and algebras of Mackey\nby Oksana Bezu schak (Kyiv Taras Shevchenko University\, Ukraine) as part of LieJor Onlin e Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nIn this talk we will discuss:\n\n1. Tensor decompositions of locally matrix algebras and their parametrization by Steinitz numbers.\n\n2. Automorphism s and derivations of locally matrix algebras.\n\n3. Automorphisms and deri vations of Mackey algebras and Mackey groups. In particular\, we describe automorphisms of all infinite simple finitary torsion groups (in the class ification of J.Hall) and derivations of all infinite-dimensional simple fi nitary Lie algebras (in the classification of A.Baranov and H.Strade).\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Aron Simis (Universidade Federal de Pernambuco\, Brazil) DTSTART;VALUE=DATE-TIME:20211028T170000Z DTEND;VALUE=DATE-TIME:20211028T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/56 DESCRIPTION:Title: Some conjectures in commutative algebra\nby Aron Simis (Univer sidade Federal de Pernambuco\, Brazil) as part of LieJor Online Seminar: A lgebras\, representations\, and applications\n\n\nAbstract\nThere are "big " conjectures and not-so-big ones in the field. Some of the first have eit her been solved (often by unexpected tools) or are still pending like a fr uit on the top of a tree with delicate branches\, making it often hard for a layperson like some of us. This talk is about more modest conjectures\, at anyone's reach and pending from trees with more stable branches. Some of these may have some interest in algebraic geometry.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/56/ END:VEVENT BEGIN:VEVENT SUMMARY:Vsevolod Gubrev (Sobolev Institute of Mathematics\, Novosibirsk\, Russia) DTSTART;VALUE=DATE-TIME:20211104T170000Z DTEND;VALUE=DATE-TIME:20211104T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/57 DESCRIPTION:Title: Embedding of Loday algebras into Rota-Baxter algebras\nby Vsev olod Gubrev (Sobolev Institute of Mathematics\, Novosibirsk\, Russia) as p art of LieJor Online Seminar: Algebras\, representations\, and application s\n\n\nAbstract\nIt is known that every Rota-Baxter algebra of weight 0 (1 ) gives rise to a prealgebra (postalgebra). In 2013\, it was proved that e very pre- or postalgebra injectively embeds into corresponding Rota-Baxter algebra of weight 0 or 1 respectively. We study the structure and the PBW -property of the universal enveloping Rota-Baxter algebra of a given pre- or post-Lie algebra.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/57/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexandre Grishkov (Universidade de São Paulo\, Brazil) DTSTART;VALUE=DATE-TIME:20211111T170000Z DTEND;VALUE=DATE-TIME:20211111T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/58 DESCRIPTION:Title: 12th Hilbert problem and Carlitz-Drinfeld-Anderson modules\nby Alexandre Grishkov (Universidade de São Paulo\, Brazil) as part of LieJo r Online Seminar: Algebras\, representations\, and applications\n\n\nAbstr act\nThe well known Kronecker-Weber theorem affirms that every finite abel ian extension of the field \\(Q\\) of rational numbers belongs to some cy clotomic extension \\(Q(t|t^n=1)\\). In his 12th problem D.Hilbert asked h ow to generalize this theorem for other global fields. In this talk\, we g ive the exposition of atual state of this problem together with the conne ction with Carlitz-Drinfeld-Anderson modules.

Recall that Anders on module \\(M\\) is a (left)module over non-commutative ring \\(R=C_p[T\, \\tau]\\)\, \\(T\\tau=\\tau T\\)\, \\(\\tau a=a^p \\tau\\)\, where \\(C_p\ \) is a some field of characteristic \\(p>0\\)\, such that \\(M\\) is free finite generated over subrings \\(C_p[T]\\) and \\(C_p\\{\\tau\\}\\).\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Marcelo Aguiar (Cornell University\, USA) DTSTART;VALUE=DATE-TIME:20211202T150000Z DTEND;VALUE=DATE-TIME:20211202T160000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/59 DESCRIPTION:Title: Lie theory relative to a hyperplane arrangement\nby Marcelo Ag uiar (Cornell University\, USA) as part of LieJor Online Seminar: Algebras \, representations\, and applications\n\n\nAbstract\nA result due to Joyal \, Klyachko\, and Stanley relates free Lie algebras to partition lattices. We will discuss the precise relationship and interpret the result in term s of the braid hyperplane arrangement. We will then extend this result to arbitrary (finite\, real\, and central) hyperplane arrangements\, and do t he same with several additional aspects of classical Hopf-Lie theory. The Tits monoid of an arrangement\, and the notion of lune\, play central role s in the discussion. This is joint work with Swapneel Mahajan.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Vladimir Sokolov (UFABC\, Brazil) DTSTART;VALUE=DATE-TIME:20211209T170000Z DTEND;VALUE=DATE-TIME:20211209T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/60 DESCRIPTION:Title: Non-Abelian Poisson brackets on projective spaces\nby Vladimir Sokolov (UFABC\, Brazil) as part of LieJor Online Seminar: Algebras\, rep resentations\, and applications\n\n\nAbstract\nWe discuss nonabelian Poiss on structures on affine and projective spaces over \\(\\mathbb{C}\\). We a lso construct a class of examples of nonabelian Poisson structures on \\(\ \mathbb{C} P^{n-1}\\) for \\(n>2\\). These nonabelian Poisson structures d epend on a modular parameter \\(\\tau\\in\\mathbb{C}\\) and an additional discrete parameter \\(k\\in\\mathbb{Z}\\)\, where \\(1\\leq k< n\\) and \\(k\,n\\) are coprime. The abelianization of these Poisson structures ca n be lifted to the quadratic elliptic Poisson algebras \\(q_{n\,k}(\\tau)\ \).\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Arturo Pianzola (University of Alberta\, Canada) DTSTART;VALUE=DATE-TIME:20211125T170000Z DTEND;VALUE=DATE-TIME:20211125T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/61 DESCRIPTION:Title: Derivations of twisted forms of Lie algebras\nby Arturo Pianzo la (University of Alberta\, Canada) as part of LieJor Online Seminar: Alge bras\, representations\, and applications\n\n\nAbstract\nThe main purpose of this talk is to explain how the theory of torsors can be used to study problems in infinite dimensional Lie theory. I will not assume that the au dience is familiar with torsors. Definitions and examples will be given. T he main application in this case is to provide a general framework (relati ve sheaves of Lie algebras) that explains/justifies a known result about t he derivations of multiloop algebras.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Iryna Kashuba (Universidade de São Paulo\, Brazil) DTSTART;VALUE=DATE-TIME:20211202T170000Z DTEND;VALUE=DATE-TIME:20211202T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/62 DESCRIPTION:Title: On the Free Jordan algebras\nby Iryna Kashuba (Universidade de São Paulo\, Brazil) as part of LieJor Online Seminar: Algebras\, represe ntations\, and applications\n\n\nAbstract\nWe will discuss a conjecture fo r the character of the homogenous components of the free Jordan algebra o n \\(d\\) generators as a \\(GL(d)\\)-module. This is joint work with Oliv ier Mathieu.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/62/ END:VEVENT BEGIN:VEVENT SUMMARY:Plamen Koshlukov (UNICAMP\, Brazil) DTSTART;VALUE=DATE-TIME:20220217T170000Z DTEND;VALUE=DATE-TIME:20220217T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/64 DESCRIPTION:Title: Gradings on upper triangular matrices\nby Plamen Koshlukov (UN ICAMP\, Brazil) as part of LieJor Online Seminar: Algebras\, representatio ns\, and applications\n\n\nAbstract\nGradings on upper triangular matrices .\; Plamen Koshlukov (UNICAMP\, Brazil)\; The upper triangular matrix alge bras are important in Linear Algebra\, and represent a powerful tool in Ri ng Theory. They also appear in the theory of PI algebras.

In addi tion to the usual associative product\, one can consider the Lie bracket a nd also the symmetric (Jordan) product on the upper triangular matrices. < br>

We discuss the group gradings on the upper triangular matrices vie wed as an associative\, Lie and Jordan algebra\, respectively. Valenti and Zaicev proved that the associative gradings are\, in a sense\, given by g radings on the matrix units. Di Vincenzo\, Valenti and Koshlukov classifie d such gradings. Later on\, Yukihide and Koshlukov\, described the Lie and the Jordan gradings. In this talk we recall some of these results as well as a new development in a rather general setting\, obtained by Yukihide a nd Koshlukov.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/64/ END:VEVENT BEGIN:VEVENT SUMMARY:Holger Petersson (FernUniversität in Hagen\, Germany) DTSTART;VALUE=DATE-TIME:20220224T170000Z DTEND;VALUE=DATE-TIME:20220224T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/65 DESCRIPTION:Title: Octonions and Albert algebras over commutative rings\nby Holge r Petersson (FernUniversität in Hagen\, Germany) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nIn the first part of the lecture\, I will focus on two properties of octonion algebras that are known to hold over fields but fail over arbitrary commu tative rings: their enumeration by means of the Cayley-Dickson constructio n\, and the norm equivalence theorem. In the second part\, I will describe a new approach to the first Tits construction of Albert algebras that\, e ven over fields\, is more general than the classical one and sheds some ne w light on the classification problem for reduced Albert algebras over com mutative rings.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/65/ END:VEVENT BEGIN:VEVENT SUMMARY:Michel Racine (Ottawa University\, Canada) DTSTART;VALUE=DATE-TIME:20220303T170000Z DTEND;VALUE=DATE-TIME:20220303T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/66 DESCRIPTION:Title: Lie Algebras afforded by Jordan algebras with particular Attention to Albert Algebras\nby Michel Racine (Ottawa University\, Canada) as part of LieJor Online Seminar: Algebras\, representations\, and applicatio ns\n\n\nAbstract\nGiven a (quadratic) Jordan algebra J over a ring k\, one obtains three Lie algebras\, the derivation algebra\, the structure algeb ra\, and the Tits algebra. We are particularly interested in the case wher e J is an Albert algebra.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/66/ END:VEVENT BEGIN:VEVENT SUMMARY:Victor Guerassimov (UFMG\, Brazil) DTSTART;VALUE=DATE-TIME:20220310T170000Z DTEND;VALUE=DATE-TIME:20220310T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/67 DESCRIPTION:Title: Random walks on groups. An introduction\nby Victor Guerassimov (UFMG\, Brazil) as part of LieJor Online Seminar: Algebras\, representati ons\, and applications\n\n\nAbstract\nGeometric methods proved to be usefu l in the study of some groups. However the geometry of the Cayley graph of a group is rather different from the geometry of classical geometric obje cts such as homogeneous spaces of Lie groups. The similarity between these two geometries grows as the scale of observation increases. And the asymp tototic behavior of them shows surprising similarity. Random walks is an e ssential tool in studying large-scale geometry of groups. On the other han d it is an interesting object for probabilists since many properties of ge neral stochastic processes are manifested here in a rather simple form. In my talk\, I will provide an elementary introduction to this vast area. No special knowledge beyond the usual university mathematics is required.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/67/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Stolin (Chalmers University of Technology\, Sweden) DTSTART;VALUE=DATE-TIME:20220317T170000Z DTEND;VALUE=DATE-TIME:20220317T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/68 DESCRIPTION:Title: 40 years of Lie bialgebras: From definition to classification\ nby Alexander Stolin (Chalmers University of Technology\, Sweden) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\ n\nAbstract\nThe history of Lie bialgebras began with the paper where the Lie bialgebras were defined: V. G. Drinfeld\, "Hamiltonian structures on L ie groups\, Lie bialgebras and the geometric meaning of the classical Yang -Baxter equations"\, Dokl. Akad. Nauk SSSR\, 268:2 (1983) Presented: L.D. Faddeev. Received: 04.06.1982.

The aim of my talk is to celebrate 40 years of Lie bialgebras in mathematics and to explain how these import ant algebraic structures can be classified. This classification goes "hand in hand" with the classification of the so-called Manin triples and Drinf eld doubles also introduced in Drinfeld's paper cited above.

The ingenious idea how to classify Drinfeld doubles associated with Lie algebr as possessing a root system is due to F. Montaner and E. Zelmanov. In part icular\, using their approach the speaker classified Lie bialgeras\, Manin triples and Drinfeld doubles associated with a simple finite dimensional Lie algebra g (the paper was based on a private communication by E. Zelman ov and it was published in Comm. Alg. in 1999).

Further\, in 201 0\, F. Montaner\, E. Zelmanov and the speaker published a paper in Selecta Math.\, where they classified Drinfeld doubles on the Lie algebra of the formal Taylor power series g[[u]] and all Lie bialgebra structures on the polynomial Lie algebra g[u].

Finally\, in March 2022 S. Maximov\ , E. Zelmanov and the speaker published an Arxive preprint\, where they ma de a crucial progress towards a complete classification of Manin triples and Lie bialgebra structures on g[[u]].

Of course\, it is impossi ble to compress a 40 years history of the subject in one talk but the spe aker will try his best to do this.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/68/ END:VEVENT BEGIN:VEVENT SUMMARY:Yuriy A. Drozd (Kiev University\, Ukraine) DTSTART;VALUE=DATE-TIME:20220324T170000Z DTEND;VALUE=DATE-TIME:20220324T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/69 DESCRIPTION:Title: Morita Theory for noncommutative varieties\nby Yuriy A. Drozd (Kiev University\, Ukraine) as part of LieJor Online Seminar: Algebras\, r epresentations\, and applications\n\n\nAbstract\nMorita theorem gives a cr iterion of equivalence of categories of modules over rings. On the other h and\, Gabriel proved that the category of coherent sheaves defines a Noeth erian scheme up to isomorphism. We have established a result which is in a sense\, a union and a combination of these two theorems. Namely\, we show that the category of coherent sheaves over a Noetherian non-commutative s cheme completely defines its center and the schemes with the same center a re Morita equivalent if and only if one of them is isomorphic to the schem e of endomorphisms of a local progeneretor of the other.

It is a comm on work with Igor Burban.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/69/ END:VEVENT BEGIN:VEVENT SUMMARY:Antonio M. Peralta (Universidad de Granada\, Spain) DTSTART;VALUE=DATE-TIME:20220331T170000Z DTEND;VALUE=DATE-TIME:20220331T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/70 DESCRIPTION:Title: How can we apply Jordan structures to reinterpret Wigner-Uhlhorn t heorem?\nby Antonio M. Peralta (Universidad de Granada\, Spain) as par t of LieJor Online Seminar: Algebras\, representations\, and applications\ n\n\nAbstract\nUp to date\, much has been written about E. Wigner and U. U hlhron theorems and their importance for physics and mathematics. For the sake of conciseness\, let us go straight to some of the starring results. There are six mathematical models employed in quantum mechanics\, among th em we have:

- The C\\(^*\\)-algebra \\(B(H)\\) of bounded operators&# 59
- The Jordan algebra \\(B(H)_{sa}\\) of bounded self-adjoint operators;
- The orthomodular lattice \\(\\mathbf{L}\\) of closed subspaces of \\(H\\)\, equivalently\, the lattice of all projection s in \\(B(H)\\)\,

Th e natural automorphisms of these mathematical models (i.e.\, the bijection s \\(f\\) on these sets preserving the corresponding relevant structure: a ssociative product and involution\, Jordan product\, and orthogonality and order between subspaces or projections) represent the symmetry groups of quantum mechanics and are endowed with natural topologies induced by the p robabilistic structure of quantum mechanics. It is known that these symmet ry groups are all isomorphic when dim\\((H)\\geq 3\\). The last restrictio n exclude rank two\, where there are no more than two orthogonal projectio ns. This equivalence can be seen as the celebrated Wigner unitary-antiunit ary theorem.

By replacing the set of projections \\(\\mathcal{P}(H) \\) by the wider set\, \\(PI(H) = \\mathcal{U}(B(H))\\)\, of all partial isometries on \\(H\\)\, L. Molná proved in [3] the following result: L et \\(H\\) be a complex Hilbert space with dim\\((H)\\geq 3\\). Suppose th at \\(\\Phi : \\mathcal{U}(B(H))\\to \\mathcal{U}(B(H))\\) is a bijective transformation which preserves the natural partial ordering and the orthog onality between partial isometries in both directions. If \\(\\Phi\\) is c ontinuous (in the operator norm) at a single element of \\(\\mathcal{U}(B( H))\\) different from \\(0\\)\, then \\(\\Phi\\) extends to a real linear triple isomorphism. %Here we consider the standard partial ordering on \\( PI(H)\\) given by \\( e\\leq u\\) if and only if \\(u-e\\) is a partial is ometry orthogonal to \\(e\\).

During this talk we shall present new results\, obtained in collaboration with Y. Friedman (see [1])\, showing that an extension of the previous results is possible in the case of a bij ection between the lattices of tripotents of two Cartan factors and atomic JBW\\(^*\\)-triples non-containing rank-one Cartan factors. These new res ult provide new models to understand the quantum models. We shall also see how the results provide new alternatives to complement recent studies by J. Hamhalter [2] proving that the set of partial isometries with its par tial order and orthogonality relation is a complete Jordan invari ant for von Neumann algebras.

[1] Y. Friedm an\, A.M. Peralta\, Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors\, to appear in

[2] J. Hamhalter\, Dye's theorem for tripotents in von Neumann algebras and JBW\\(^*\\)-triples\,

[3] L. Molnár\, On certain automorphisms of sets of partial isometries\,

[4] U. Uhlhorn\, Representati on of symmetry transformations in quantum mechanics\,

[5] E.P. Wigner\,

If \\(K\\) is a subgroup o f a finite group \\(G\\)\, the probability that an element of \\(G\\) comm utes with an element of \\(K\\) is denoted by \\(Pr(K\,G)\\). The probabil ity that two randomly chosen elements of \\(G\\) commute is denoted by \\( Pr(G)\\). A well known theorem\, due to P. M. Neumann\, says that if \\(G\ \) is a finite group such that \\(Pr(G)\\geq\\epsilon\\)\, then \\(G\\) ha s a nilpotent normal subgroup \\(T\\) of class at most \\(2\\) such that b oth the index \\([G:T]\\) and the order \\(|[T\,T]|\\) are \\(\\epsilon\\) -bounded.

In the talk we will discuss a stronger version of Neum ann's theorem: if \\(K\\) is a subgroup of \\(G\\) such that \\(Pr(K\,G)\\ geq\\epsilon\\)\, then there is a normal subgroup \\(T\\leq G\\) and a sub group \\(B\\leq K\\) such that the indexes \\([G:T]\\) and \\([K:B]\\) and the order of the commutator subgroup \\([T\,B]\\) are \\(\\epsilon\\)-bou nded.

We will also discuss a number of corollaries of this resul t. A typical application is that if in the above theorem \\(K\\) is the ge neralized Fitting subgroup \\(F^*(G)\\)\, then \\(G\\) has a class-2-nilpo tent normal subgroup \\(R\\) such that both the index \\([G:R]\\) and the order of the commutator subgroup \\([R\,R]\\) are \\(\\epsilon\\)-bounded. \n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/71/ END:VEVENT BEGIN:VEVENT SUMMARY:Eduardo do Nascimento Marcos (IME-USP\, Brazil) DTSTART;VALUE=DATE-TIME:20220414T170000Z DTEND;VALUE=DATE-TIME:20220414T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/72 DESCRIPTION:Title: Koszul e homogeneous triples for algebras with two relations\n by Eduardo do Nascimento Marcos (IME-USP\, Brazil) as part of LieJor Onlin e Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nTh is talk is based on a joint work with Yury Volkov. We define the category of homogeneous triples\, which is equivalent to the category of graded alg ebras\, with a fixed semisimple degree zero part. We apply the results to algebras whose defining ideal has two generators\, and give a partial clas sification.

We thank Fapesp\, grant 2018/23690-6\, for the suppo rt.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/72/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexei Miasnikov (Stevens Institute of Technology\, USA) DTSTART;VALUE=DATE-TIME:20220421T170000Z DTEND;VALUE=DATE-TIME:20220421T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/73 DESCRIPTION:Title: Rich groups and weak second order logic\nby Alexei Miasnikov ( Stevens Institute of Technology\, USA) as part of LieJor Online Seminar: A lgebras\, representations\, and applications\n\n\nAbstract\n"What can one describe by first-order formulas in a given group A?" - is an old and inte resting question. Of course\, this depends on the group A. For example\, i n a free group only cyclic subgroups (and the group itself) are definable in the first-order logic\, but in a free monoid of finite rank any finitel y generated submonoid is definable. A group A is called rich if the first- order logic in A is equivalent to the weak second order logic. Surprisingl y\, there are a lot of interesting groups\, rings\, semigroups\, etc.\, wh ich are rich. I will describe various algebraic\, geometric\, and algorith mic properties that are first-order definable in rich groups and apply the se to some open problems. Weak second order logic can be introduced into a lgebraic structures in different ways: via HF-logic\, or list superstructu res over A\, or computably enumerable infinite disjunctions and conjunctio ns\, or via finite binary predicates\, etc. I will describe a particular f orm of this logic which is especially convenient to use in algebra and sho w how to effectively translate such weak second order formulas into the eq uivalent first-order ones in the case of a rich group A.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/73/ END:VEVENT BEGIN:VEVENT SUMMARY:Mark Sapir (Vanderbilt University\, USA) DTSTART;VALUE=DATE-TIME:20220428T170000Z DTEND;VALUE=DATE-TIME:20220428T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/74 DESCRIPTION:Title: Subgroups of the R.Thompson group F\nby Mark Sapir (Vanderbilt University\, USA) as part of LieJor Online Seminar: Algebras\, representa tions\, and applications\n\n\nAbstract\nThis is joint work with Gili Golan -Polak. We describe the so-called closed subgroups of F. In particular\, w e construct a subgroup of F with easily decidable membership problem and u ndecidable conjugacy problem\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/74/ END:VEVENT BEGIN:VEVENT SUMMARY:Ivan Penkov (Jacobs University Bremen\, Germany) DTSTART;VALUE=DATE-TIME:20220505T170000Z DTEND;VALUE=DATE-TIME:20220505T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/75 DESCRIPTION:Title: New analogues of category O for the Lie algebra \\(sl(\\infty)\\)< /a>\nby Ivan Penkov (Jacobs University Bremen\, Germany) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstra ct\nI will recall several highest weight categories for \\(sl(\\infty)\\) studied in the past decade\, and will then report on the newest highest we ight categories introduced by P. Zadunaisky. A main point is the use a non -obvious Borel subalgebra plus a semi-large annihilator condition. As a si de effect\, the new categories produce interesting and challenging combina torics.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/75/ END:VEVENT BEGIN:VEVENT SUMMARY:Yuri Bahturin (Memorial University of Newfoundland\, Canada) DTSTART;VALUE=DATE-TIME:20220512T170000Z DTEND;VALUE=DATE-TIME:20220512T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/76 DESCRIPTION:Title: Group Gradings and Actions of Pointed Hopf Algebras\nby Yuri B ahturin (Memorial University of Newfoundland\, Canada) as part of LieJor O nline Seminar: Algebras\, representations\, and applications\n\n\nAbstract \nPointed Hopf algebras are a wide class of Hopf algebras\, including grou p algebras and enveloping algebras of Lie algebras. In this talk\, based o n a recent work with Susan Montgomery\, we study actions of pointed Hopf a lgebras on simple algebras. These actions are known to be inner\, as in th e case of Skolem - Noether theorem. We try to give explicit descriptions\, whenever possible\, and consider Taft algebras\, their Drinfeld doubles a nd some quantum groups.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/76/ END:VEVENT BEGIN:VEVENT SUMMARY:José Oswaldo Lezama Serrano (Universidad Nacional de Colombia\, C olombia) DTSTART;VALUE=DATE-TIME:20220519T170000Z DTEND;VALUE=DATE-TIME:20220519T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/77 DESCRIPTION:Title: Algebraic sets\, ideals of points and the Hilbert's Nullstellensat z theorem for skew PBW extensions\nby José Oswaldo Lezama Serrano (Un iversidad Nacional de Colombia\, Colombia) as part of LieJor Online Semina r: Algebras\, representations\, and applications\n\n\nAbstract\nIn this ta lk we define the algebraic sets and the ideal of points for bijective skew PBW extensions with coefficients in left Noetherian domains. Some propert ies of affine algebraic sets of commutative algebraic geometry will be ext ended\, in particular\, a Zariski topology will be constructed. Assuming a dditionally that the extension is quasi-commutative with polynomial center and the ring of coefficients is an algebraically closed field\, we will p rove an adapted version of Hilbert's Nullstellensatz theorem that covers t he classical one. The Gröbner bases of skew PBW extensions will be used f or defining the algebraic sets and for proving the main theorem. Many key algebras and rings coming from mathematical physics and non-commutative al gebraic geometry are skew PBW extensions.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/77/ END:VEVENT BEGIN:VEVENT SUMMARY:Efim Zelmanov (University of California\, San Diego\, USA) DTSTART;VALUE=DATE-TIME:20220526T170000Z DTEND;VALUE=DATE-TIME:20220526T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/78 DESCRIPTION:by Efim Zelmanov (University of California\, San Diego\, USA) as part of LieJor Online Seminar: Algebras\, representations\, and applica tions\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/78/ END:VEVENT BEGIN:VEVENT SUMMARY:David A. Jordan (Sheffield University\, UK) DTSTART;VALUE=DATE-TIME:20220602T170000Z DTEND;VALUE=DATE-TIME:20220602T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/79 DESCRIPTION:Title: Skew derivations of quantum spaces\nby David A. Jordan (Sheffi eld University\, UK) as part of LieJor Online Seminar: Algebras\, represen tations\, and applications\n\n\nAbstract\nLet $n$ be a positive integer an d let $Q = (q_{ij})$ be a multipicatively antisymmetric $n \\times n$ ma trix\nover a field $\\mathbb{K}$\, that is $q_{ii}=1$ for $1\\leq i\\leq n $ and\, for $1\\leq i\,j\\leq n$\, $q_{ij}\\neq 0$ and $q_{ji}=q_{ij}^{-1 }$. \nThe

This is joint work with A. Kannan and V. Ostrik.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/81/ END:VEVENT BEGIN:VEVENT SUMMARY:Misha Dokuchaev (IME-USP (Brazil)) DTSTART;VALUE=DATE-TIME:20220623T170000Z DTEND;VALUE=DATE-TIME:20220623T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/82 DESCRIPTION:Title: Strong equivalence of graded algebras\nby Misha Dokuchaev (IME -USP (Brazil)) as part of LieJor Online Seminar: Algebras\, representation s\, and applications\n\n\nAbstract\nWe introduce the notion of a strong eq uivalence between graded algebras and prove that any partially-strongly-gr aded algebra by a group G is strongly-graded-equivalent to the skew group algebra by a product partial action of G. We show that strongly-graded-equ ivalence preserves strong gradings and is nicely related to Morita equival ence of product partial actions. Furthermore\, we show that strongly-grade d-equivalent partially-strongly-graded algebras with orthogonal local unit s are stably isomorphic as graded algebras. This is a part of a joint prep rint with Fernando Abadie and Ruy Exel.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/82/ END:VEVENT BEGIN:VEVENT SUMMARY:Sergio Lopez-Permouth (Ohio University\, USA) DTSTART;VALUE=DATE-TIME:20220630T170000Z DTEND;VALUE=DATE-TIME:20220630T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/83 DESCRIPTION:Title: On the isomorphism problem for basic modules\nby Sergio Lopez- Permouth (Ohio University\, USA) as part of LieJor Online Seminar: Algebra s\, representations\, and applications\n\n\nAbstract\nWhile mutual congeni ality of bases has been known to guarantee that basic modules from so rela ted bases are isomorphic\, the question of what can be said about isomorph ism of basic modules in general has remained open. We show that neither of two possible extremes must hold. For some algebras\, it is possible\, for basic modules to be non-isomorphic. Also\, it is possible\, for some alg ebras\, that all basic modules be isomorphic.

We show that there a re at least as many pairwise non-isomorphic basic modules over the \\(F\\) -algebra \\(F[x]\\) of polynomials in a single variable as there are eleme nts in \\(F\\). We show that basic modules over \\(F[x]\\) can be non-iso morphic when they are induced by discordant bases and also even when there is a (non-mutual) congeniality among them. In the process and as a byprod uct\, we introduce the notion of domains of divisibility of modules over a rbitrary rings and explore some of the properties of a divisibility profil e.

At the opposite end of the spectrum\, we present an algebra whe re all basic modules are isomorphic\, regardless of congeniality.

This is a report on joint work with: C. Arellano\, P. Aydogdu\, R. Muhamma d\, and M. Zailaee.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/83/ END:VEVENT BEGIN:VEVENT SUMMARY:Ivan Ezequiel Angiono (National University of Cordoba\, Argentina) DTSTART;VALUE=DATE-TIME:20220707T170000Z DTEND;VALUE=DATE-TIME:20220707T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/84 DESCRIPTION:Title: Finite-dimensional pointed Hopf algebras over central extensions o f abelian groups\nby Ivan Ezequiel Angiono (National University of Cor doba\, Argentina) as part of LieJor Online Seminar: Algebras\, representat ions\, and applications\n\n\nAbstract\nOne of the most studied kinds of fi nite-dimensional Hopf algebras is the family of pointed ones: it means tha t the coradical is the algebra of the group-like elements. When the group is abelian\, all such examples are known following the so-called Lifting M ethod by Andruskiewitsch-Schneider and include deformations of small quant um groups\, their super analogues and some exceptional examples of Nichols algebras. When the group is not abelian\, the classification is not known yet. Even more\, the first step of the Lifting Method (the computation of all finite-dimensional Nichols algebras) has not been completed: the clas sification has been performed by Heckenberger-Vendramin when the elements in degree one form a non-simple Yetter-Drinfeld module\, and consist of lo w rank exceptions and large rank families.\n\nIn this talk we will present finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associate d with a simple Lie algebra together with a Dynkin diagram automorphism.\n \nWe will show conversely that every finite-dimensional pointed Hopf algeb ra over a non-abelian group with a non-simple infinitesimal braiding is of this form for large rank families. The proof follows the steps of the Lif ting Method. Indeed we prove that the large rank families are cocycle twis ts of Nichols algebras constructed by Lentner as foldings of Nichols algeb ras of Cartan type over abelian groups by outer automorphisms. This enable s us to give uniform Lie-theoretic descriptions of the large rank families \, prove generation in degree one and construct liftings.\n\nWe also show that every lifting is a cocycle deformation of the corresponding coradical ly graded Hopf algebra using an explicit presentation by generators and re lations of the Nichols algebra.\n\nThe talk is based on a joint work with Simon Lentner and Guillermo Sanmarco.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/84/ END:VEVENT BEGIN:VEVENT SUMMARY:Jason Bell (University of Waterloo\, Canada) DTSTART;VALUE=DATE-TIME:20220728T170000Z DTEND;VALUE=DATE-TIME:20220728T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/85 DESCRIPTION:Title: Recent results on the Dixmier-Moeglin equivalence\nby Jason Be ll (University of Waterloo\, Canada) as part of LieJor Online Seminar: Alg ebras\, representations\, and applications\n\n\nAbstract\nDixmier and Moeg lin showed that if \\(L\\) is a finite-dimensional complex Lie algebra the n the primitive ideals of the enveloping algebra \\(U(L)\\) are the prime ideals of \\({\\rm Spec}(U(L))\\) that are locally closed in the Zariski t opology. In addition\, they proved that a prime ideal \\(P\\) of \\(U(L)\\ ) is primitive if and only if the Goldie ring of quotients of \\(U(L)/P\\) has the property that its centre is just the base field of the complex nu mbers. Algebras that share this characterization of primitive ideals are s aid to satisfy the Dixmier-Moeglin equivalence. We give an overview of th is property and mention some recent work on proving this equivalence holds for certain classes of twisted homogenous coordinate rings and classes of Hopf algebras of small Gelfand-Kirillov dimension.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/85/ END:VEVENT BEGIN:VEVENT SUMMARY:Ruy Exel (UFSC\, Brazil) DTSTART;VALUE=DATE-TIME:20220804T170000Z DTEND;VALUE=DATE-TIME:20220804T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/86 DESCRIPTION:Title: The opaque ideal\nby Ruy Exel (UFSC\, Brazil) as part of LieJo r Online Seminar: Algebras\, representations\, and applications\n\n\nAbstr act\nGiven a C*-algebra \\(B\\)\, and a regular\, abelian\, sub-C*-algebra \\(A\\subseteq B\\)\, we will discuss the

- joint results wi th Gregor Kemper and Fabian Reimers on separating invariants for the ring of multisymmetric polynomials in \\(m\\) sets of \\(n\\) variables over an arbitrary field \\(\\mathbb{F}\\);
- joint results with Alexander Z ubkov on separating invariants of several octonions with respect to the ac tion of \\(G_2\\);
- joint results with Felipe Barbosa Cavalcante on
separating invariants of \\(2\\times 2\\) and \\(3\\times 3\\) matrices.<
/ol>\n
LOCATION:https://researchseminars.org/talk/LieJor_Seminar/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Petrogradsky (UnB\, Brazil)
DTSTART;VALUE=DATE-TIME:20220825T170000Z
DTEND;VALUE=DATE-TIME:20220825T180000Z
DTSTAMP;VALUE=DATE-TIME:20221209T220251Z
UID:LieJor_Seminar/90
DESCRIPTION:Title: Growth in Lie algebras\nby Victor Petrogradsky (UnB\, Brazil)
as part of LieJor Online Seminar: Algebras\, representations\, and applica
tions\n\n\nAbstract\nDifferent versions of Burnside Problem ask what one can say about finitely generated
periodic groups under additional assumptions. For associative algebras\,
Kurosh type problems ask simila
r questions about properties of finitely generated nil (more generally\, a
lgebraic) algebras. Similarly\, one considers finitely generated restricte
d Lie algebras with a nil \\(p\\)-mapping. Now we study an oscillating intermediate growth in nil restricted Lie algebras.

Name ly\, for any field of positive characteristic\, we construct a family of 3 -generated restricted Lie algebras of intermediate oscillating growth. We call them*Phoenix algebras*\, because of the following.

a) For in finitely many periods of time the algebra is "almost dying" by having a < i>quasi-linear growth\, namely the lower Gelfand-Kirillov dimension is one\, more precisely\, he growth is of type \\(n \\big(\\underbrace{\\ln \\cdots \\ln}_{q\\ \\text{times}} n\\big )^{\\kappa}\\)\, where \\(q\\in\ \mathbb N\\)\, \\(\\kappa>0\\) are constants.

b) On the other hand\, fo r infinitely many \\(n\\) the growth function has a rather fast intermedia te behaviour of type \\(\\exp( n/ (\\ln n)^{\\lambda})\\)\, \\(\\lambda\\) being a constant determined by characteristic\, for such periods the alge bra is "resuscitating".

c) Moreover\, the growth function is bounded an d oscillating between these two types of behaviour.

d) These restricted Lie algebras have a nil \\(p\\)-mapping.

We also construct nil Lie superalgebras and nil Jordan superalgebras of similar oscillating interme diary growth over arbitrary field.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/90/ END:VEVENT BEGIN:VEVENT SUMMARY:Alistair Savage (University of Ottawa\, Canada) DTSTART;VALUE=DATE-TIME:20220901T170000Z DTEND;VALUE=DATE-TIME:20220901T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/91 DESCRIPTION:Title: Diagratification\nby Alistair Savage (University of Ottawa\, C anada) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nWe will explain how one can construct diagramm atic presentations of categories of representations of Lie groups and thei r associated quantum groups using only a small amount of information about these categories. To illustrate the technique in concrete terms\, we will focus on the exceptional Lie group of type F4.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/91/ END:VEVENT BEGIN:VEVENT SUMMARY:Allan Berele (De Paul University\, USA) DTSTART;VALUE=DATE-TIME:20220922T170000Z DTEND;VALUE=DATE-TIME:20220922T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/92 DESCRIPTION:Title: Poincaré Series of the Trace Rings of Generic Matrices\nby Al lan Berele (De Paul University\, USA) as part of LieJor Online Seminar: Al gebras\, representations\, and applications\n\n\nAbstract\nWe first give s ome background on the Poincare series of the algebra of generic matrices a nd its associated trace ring\, and then focus on some recent work\, includ ing a conjecture for the denominator of the one variable series for the tr ace rings. Time permitting we will also say a bit about traces of direct s ums.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/92/ END:VEVENT BEGIN:VEVENT SUMMARY:Alberto Facchini (Università degli Studi di Padova\, Italia) DTSTART;VALUE=DATE-TIME:20220929T170000Z DTEND;VALUE=DATE-TIME:20220929T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/93 DESCRIPTION:Title: Multiplicative lattices\, skew braces\nby Alberto Facchini (Un iversità degli Studi di Padova\, Italia) as part of LieJor Online Seminar : Algebras\, representations\, and applications\n\n\nAbstract\nThe multipl icative lattices we will consider are those defined in the paper [3]\, pub lished in February 2022. Multiplicative lattices yield the natural setting in which several basic mathematical questions concerning algebraic struct ures find their answer (Zariski spectrum\, nilpotency\, solvability\, abel ian algebraic structures\,...) We will consider the particular case of ske w braces\, which appear in connection to the study of the Yang-Baxter equa tion ([2]\, [3] and [4]).\n\n[1] D. Bourn\, A. Facchini and M. Pompili\, A spects of the Category SKB of Skew Braces\, submitted for publication\, av ailable in arXiv\, 2022\n\n[2] A. Facchini\, Algebraic structures from the point of view of complete multiplicative lattices\, accepted for publicat ion in ``Rings\, Quadratic Forms\, and their Applications in Coding Theory ''\, Contemporary Math.\, 2022\, available at: http://arxiv.org/abs/2201.0 3295\n\n[3] A. Facchini\, C. A. Finocchiaro and G. Janelidze\, Abstractly constructed prime spectra\, Algebra universalis 83(1) (2022).\n\n[4] A. Fa cchini\, F. de Giovanni and M. Trombetti\, Spectra of Groups\, Algebras Re p. Theory\, Online first articles published 5 June 2022.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/93/ END:VEVENT BEGIN:VEVENT SUMMARY:Diogo Diniz (Universidade Federal de Campina Grande\, Brazil) DTSTART;VALUE=DATE-TIME:20221117T170000Z DTEND;VALUE=DATE-TIME:20221117T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/94 DESCRIPTION:Title: Gradings on block-triangular matrix algebras\nby Diogo Diniz ( Universidade Federal de Campina Grande\, Brazil) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nUppe r triangular\, and more generally\, block-triangular matrices\, are rather important in Linear Algebra\, and also in Ring theory\, namely in the the ory of PI algebras. The group gradings on such algebras have been studied extensively during the last decades. In 2007 A. Valenti and M. Zaicev con jectured that every grading on these algebras is obtained from an elementa ry grading on a block-triangular matrix algebra and a division grading on a matrix algebra. In this talk we present recent results on this problem.\ n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/94/ END:VEVENT BEGIN:VEVENT SUMMARY:Sergey Pchelintsev and Oleg Shashkov (Financial University under t he Government of the Russian Federation\, Russia) DTSTART;VALUE=DATE-TIME:20220915T170000Z DTEND;VALUE=DATE-TIME:20220915T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/95 DESCRIPTION:Title: Simple right-alternative superalgebras\nby Sergey Pchelintsev and Oleg Shashkov (Financial University under the Government of the Russia n Federation\, Russia) as part of LieJor Online Seminar: Algebras\, repres entations\, and applications\n\n\nAbstract\nWe are going to talk about wha t is known about simple right-alternative superalgebras at this time. Righ t alternative superalgebras can be divided into two classes\, these are un ital and non-unital superalgebras. In the unital case\, the case of simple superalgebras with a semisimple even part is completely described. In the non-unital case\, we describe a class of simple superalgebras with zero m ultiplication of the even part\, which we call the class of singular super algebras. A scheme of the so-called extended double is given and it is pro ved that every singular superalgebra is an extended double. The dimensions for which there are no singular superalgebras are indicated\, and example s of singular superalgebras of all other dimensions are given.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/95/ END:VEVENT BEGIN:VEVENT SUMMARY:Nikolay Nikolov (Oxford University\, UK) DTSTART;VALUE=DATE-TIME:20221006T170000Z DTEND;VALUE=DATE-TIME:20221006T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/96 DESCRIPTION:Title: On conjugacy classes of profinite groups\nby Nikolay Nikolov ( Oxford University\, UK) as part of LieJor Online Seminar: Algebras\, repre sentations\, and applications\n\n\nAbstract\nIt is well-known that the num ber of conjugacy classes of a finite group G tends to infinity as the size of G tends to infinity. There is no such result for a general infinite gr oup. In this talk I will discuss the situation when G is a profinite group and show that the number of conjugacy of G is then uncountable unless G i s finite. The proof depends on many classical results on finite groups and in particular the classification of the finite simple groups. This is joi nt work with Andrei Jaikin-Zapirain.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/96/ END:VEVENT BEGIN:VEVENT SUMMARY:Carolina Araujo (IMPA\, Brazil) DTSTART;VALUE=DATE-TIME:20221013T170000Z DTEND;VALUE=DATE-TIME:20221013T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/97 DESCRIPTION:Title: Higher Fano manifolds\nby Carolina Araujo (IMPA\, Brazil) as p art of LieJor Online Seminar: Algebras\, representations\, and application s\n\n\nAbstract\nFano manifolds are complex projective manifolds having po sitive first Chern class. The positivity condition on the first Chern clas s has far reaching geometric and arithmetic implications. For instance\, F ano manifolds are covered by rational curves\, and families of Fano manifo lds over one dimensional bases always admit holomorphic sections. In recen t years\, there has been great effort towards defining suitable higher ana logues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance\, they should be covered by higher dimensional rational varietie s\, and families of higher Fano manifolds over higher dimensional bases sh ould admit meromorphic sections (modulo Brauer obstruction). In this talk\ , I will discuss a possible notion of higher Fano manifolds in terms of po sitivity of higher Chern characters\, and describe special geometric featu res of these manifolds.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/97/ END:VEVENT BEGIN:VEVENT SUMMARY:Eduardo Esteves (IMPA\, Brazil) DTSTART;VALUE=DATE-TIME:20221020T170000Z DTEND;VALUE=DATE-TIME:20221020T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/98 DESCRIPTION:Title: Quiver representations arising from degenerations of linear series \nby Eduardo Esteves (IMPA\, Brazil) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nWe describe all the schematic limits of divisors associated to any family of linear se ries on any one-dimensional family of projective varieties degenerating to any connected reduced projective scheme X defined over any field\, under the assumption that the total space of the family is regular along X. More precisely\, the degenerating family gives rise to a special quiver Q\, ca lled a Z^n-quiver\, a special representation L of Q in the category of lin e bundles over X\, called a maximal exact linked net\, and a special subre presentation V of the representation induced from L by taking global secti ons\, called a pure exact finitely generated linked net. Given g=(Q\, L\, V) satisfying these properties\, we prove that the quiver Grassmanian G of subrepresentations of V of pure dimension 1\, called a linked projective space\, is Cohen-Macaulay\, reduced and of pure dimension. Furthermore\, w e prove that there is a morphism from G to the Hilbert scheme of X whose i mage parameterizes all the schematic limits of divisors along the degenera ting family of linear series if g arises from one. Joint work with Eduardo Vital and Renan Santos.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/98/ END:VEVENT BEGIN:VEVENT SUMMARY:Laurent Bartholdi (Saarland University\, Germany) DTSTART;VALUE=DATE-TIME:20221027T170000Z DTEND;VALUE=DATE-TIME:20221027T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/99 DESCRIPTION:Title: Dimension series and homotopy groups of spheres\nby Laurent Ba rtholdi (Saarland University\, Germany) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nThe lower cen tral series of a group \\(G\\) is defined by \\(\\gamma_1=G\\) and \\(\\ga mma_n = [G\,\\gamma_{n-1}]\\). The "dimension series"\, introduced by Magn us\, is defined using the group algebra over the integers: \\(\\delta_n = \\{g: g-1\\text{ belongs to the \\(n\\)-th power of the augmentation ideal }\\}\\).

It has been\, for the last 80 years\, a fundamental proble m of group theory to relate these two series. One always has \\(\\delta_n\ \ge\\gamma_n\\)\, and a conjecture by Magnus\, with false proofs by Cohn\, Losey\, etc.\, claims that they coincide; but Rips constructed an exam ple with \\(\\delta_4/\\gamma_4\\) cyclic of order 2. On the positive side \, Sjogren showed that \\(\\delta_n/\\gamma_n\\) is always a torsion group \, of exponent bounded by a function of \\(n\\). Furthermore\, it was beli eved (and falsely proven by Gupta) that only \\(2\\)-torsion may occur.

In joint work with Roman Mikhailov\, we prove however that every tors ion abelian group may occur as a quotient \\(\\delta_n/\\gamma_n\\); th is proves that Sjogren's result is essentially optimal.

Even more i nterestingly\, we show that this problem is intimately connected to the ho motopy groups \\(\\pi_n(S^m)\\) of spheres; more precisely\, the quotie nt \\(\\delta_n/\\gamma_n\\) is related to the difference between homotopy and homology. We may explicitly produce \\(p\\)-torsion elements starting from the order-\\(p\\) element in the homotopy group \\(\\pi_{2p}(S^2)\\) due to Serre.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/99/ END:VEVENT BEGIN:VEVENT SUMMARY:Viviane Ribeiro Tomaz da Silva (UFMG\, Brazil) DTSTART;VALUE=DATE-TIME:20221103T170000Z DTEND;VALUE=DATE-TIME:20221103T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/100 DESCRIPTION:Title: On the minimal varieties of PI *-superalgebras and the factorabil ity of their T-ideals\nby Viviane Ribeiro Tomaz da Silva (UFMG\, Brazi l) as part of LieJor Online Seminar: Algebras\, representations\, and appl ications\n\n\nAbstract\nIn this talk\, we deal with varieties of PI-supera lgebras with graded involution of finite basic rank over a field of charac teristic zero and we present some recent results concerning the minimality of these varieties (of fixed *-graded exponent) and the factorability of their *-graded polynomial identities.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/100/ END:VEVENT BEGIN:VEVENT SUMMARY:Igor Klep (University of Ljubljana\, Slovenia) DTSTART;VALUE=DATE-TIME:20221110T170000Z DTEND;VALUE=DATE-TIME:20221110T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/101 DESCRIPTION:Title: Factorization of noncommutative polynomials and Nullstellensä tze for the free algebra\nby Igor Klep (University of Ljubljana\, Slov enia) as part of LieJor Online Seminar: Algebras\, representations\, and a pplications\n\n\nAbstract\nThe singularity set of a noncommutative polynom ial \\(f=f(x_1\,\\dots\,x_d)\\) is the graded set \\(Z(f)=(Z_n(f))_n\\)\, where \\(Z_n(f)=\\{X \\in M_n^d: \\det f(X) = 0\\}.\\) Two main results wi ll be presented. Firstly\, irreducible factors of \\(f\\) are shown to be in a natural bijective correspondence with irreducible components of \\(Z_ n(f)\\) for every sufficiently large \\(n\\). In particular\, \\(f\\) is i rreducible if and only if \\(Z_n(f)\\) is eventually irreducible. Secondly \, we give Nullstellensätze for noncommutative polynomials. For instan ce\, given two noncommutative polynomials \\(f_1\,f_2\\)\, we have \\(Z(f_ 1) \\subset Z(f_2)\\) if and only if each irreducible factor of \\(f_1\\) is (up to stable associativity) an irreducible factor of \\(f_2\\). Along the way an algorithm for factorization of noncommutative polynomials will be presented.

The talk is based on joint works with Jurij Vol 9ič and Bill Helton.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/101/ END:VEVENT BEGIN:VEVENT SUMMARY:Bojko Bakalov (North Carolina State University\, USA) DTSTART;VALUE=DATE-TIME:20221208T170000Z DTEND;VALUE=DATE-TIME:20221208T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/102 DESCRIPTION:Title: An operadic approach to vertex algebras and Poisson vertex algebr as\nby Bojko Bakalov (North Carolina State University\, USA) as part o f LieJor Online Seminar: Algebras\, representations\, and applications\n\n \nAbstract\nI will start by reviewing the notions of vertex algebra\, Pois son vertex algebra\, and Lie conformal algebra\, and their relations to ea ch other. Then I will present a unified approach to all these algebras as Lie algebras in certain pseudo-tensor categories\, or equivalently\, as mo rphisms from the Lie operad to certain operads. As an application\, I will introduce a cohomology theory of vertex algebras similarly to Lie algebra cohomology\, and will show how it relates to the cohomology of Poisson ve rtex algebras and of Lie conformal algebras. The talk is based on joint wo rk with Alberto De Sole\, Reimundo Heluani\, Victor Kac\, and Veronica Vig noli.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/102/ END:VEVENT BEGIN:VEVENT SUMMARY:Matyas Domokos (Renyu Institute of Mathematics\, Budapest\, Hungar y) DTSTART;VALUE=DATE-TIME:20221124T170000Z DTEND;VALUE=DATE-TIME:20221124T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/103 DESCRIPTION:Title: Improvements of the Noether bound for polynomial invariants of fi nite groups\nby Matyas Domokos (Renyu Institute of Mathematics\, Budap est\, Hungary) as part of LieJor Online Seminar: Algebras\, representation s\, and applications\n\n\nAbstract\nGiven a field and a finite group G\, t he Noether number of G is defined as the minimal positive integer d such t hat for any finite dimensional G-module V\, the algebra of G-invariant pol ynomial functions on V is generated by elements of degree at most d. In th e talk we shall survey results (obtained mostly together with Kálmán Czi szter) on the Noether number of various finite groups.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/103/ END:VEVENT BEGIN:VEVENT SUMMARY:Pavel Zalesski (UnB\, Brazil) DTSTART;VALUE=DATE-TIME:20221215T170000Z DTEND;VALUE=DATE-TIME:20221215T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/104 DESCRIPTION:Title: Combinatorial theory of pro-p groups\nby Pavel Zalesski (UnB\ , Brazil) as part of LieJor Online Seminar: Algebras\, representations\, a nd applications\n\n\nAbstract\nFree product with amalgamation and HNN-exte nsion are two main constructions of combinatorial group theory. I shall di scuss these two constructions in the category of pro-\\(p\\) groups\, pres enting results on splittings of pro-\\(p\\) groups as an amalgamated free pro-\\(p\\) product or a pro-\\(p\\) HNN-extension and relating them with pro-\\(p\\) version of Bass-Serre's theory of groups acting on trees. I s hall also compare the pro-\\(p\\) results with similar results for abstra ct groups.\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/104/ END:VEVENT BEGIN:VEVENT SUMMARY:Felipe Yasumura (IME-USP\, Brazil) DTSTART;VALUE=DATE-TIME:20221201T170000Z DTEND;VALUE=DATE-TIME:20221201T180000Z DTSTAMP;VALUE=DATE-TIME:20221209T220251Z UID:LieJor_Seminar/105 DESCRIPTION:Title: Group gradings on the infinite dimensional triangular algebra \nby Felipe Yasumura (IME-USP\, Brazil) as part of LieJor Online Seminar: Algebras\, representations\, and applications\n\n\nAbstract\nIn the last d ecades\, there has been an increasing interest in the classification of is omorphism classes of group gradings on a given algebra. We discuss some di fficulties concerning the study of group gradings on infinite-dimensional algebras. Then\, we present our results on the classification of the gradi ngs on the infinite-dimensional triangular algebra. This is joint work wit h Waldeck Schutzer (UFSCar).\n LOCATION:https://researchseminars.org/talk/LieJor_Seminar/105/ END:VEVENT END:VCALENDAR