BEGIN:VCALENDAR
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PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Chelsea Walton (Rice University\, USA)
DTSTART:20210330T160000Z
DTEND:20210330T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/1
DESCRIPTION:Title: Un
iversal Quantum Semigrupoids\nby Chelsea Walton (Rice University\, USA
) as part of ONCAS Online Noncommutative Algebra Seminar\n\n\nAbstract\nIn
a recent paper (https://arxiv.org/abs/2008.00606)\, Hongdi Huang\, Elizab
eth Wicks\, Robert Won\, and I introduce the concept of a universal quantu
m linear semigroupoid (UQSGd). This is a weak bialgebra that coacts on a (
not necessarily connected) graded algebra A. Our main result is that when
A is the path algebra kQ of a finite quiver Q each of the various UQSGds i
ntroduced in our work is isomorphic to the face algebra attached to Q (an
important weak bialgebra due to Hayashi). Most of the talk will be dedicat
ed to setting up context and terminology towards the main result.\n
LOCATION:https://researchseminars.org/talk/ONCAS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:José Gómez Torrecillas (Universidad de Granada\, Spain)
DTSTART:20210413T160000Z
DTEND:20210413T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/2
DESCRIPTION:Title: Bi
separable extensions are not necessarily Frobenius\nby José Gómez To
rrecillas (Universidad de Granada\, Spain) as part of ONCAS Online Noncomm
utative Algebra Seminar\n\n\nAbstract\nNecessary and sufficient conditions
are given on an Ore extension A[x\; σ\, δ]\, where A is a finite dimens
ional algebra over a field F\, for being a Frobenius extension of the ring
of commutative polynomials F[x]. As a consequence\, as the title of this
talk highlights\, we provide a negative answer to a problem stated by Caen
epeel and Kadison. The involved ring-theoretical notions will be also disc
ussed. \n\n\nThe talk is based on the joint paper with F.J. Lobillo\, G.
Navarro and P. Sánchez-Hernández\n
LOCATION:https://researchseminars.org/talk/ONCAS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Marsinkovsky (Northeastern University\, USA)
DTSTART:20210427T160000Z
DTEND:20210427T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/3
DESCRIPTION:Title: On
Rings and Functors: Pictures at an Exhibition\nby Alexander Marsinkov
sky (Northeastern University\, USA) as part of ONCAS Online Noncommutative
Algebra Seminar\n\n\nAbstract\nIt is banal to say that rings reveal their
nature through their module categories. However\, in the last few years w
e have seen indications that a similar statement can be made about stable
module categories. Those are categories whose objects are the usual module
s\, but morphisms are formed by modding out maps factoring through project
ives (or injectives). Additive functors defined on such categories are als
o said to be stable\, and the study of such functors adds even more insigh
ts. The goal of this talk is to illustrate this thesis with simple example
s. We shall see how several familiar classes of rings can be characterized
by properties of their stable functors. This will be an expository talk.
No prior knowledge of functor categories is assumed or needed.\n
LOCATION:https://researchseminars.org/talk/ONCAS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivo Herzog (Ohio State University\, USA)
DTSTART:20210525T160000Z
DTEND:20210525T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/5
DESCRIPTION:Title: Th
e Positselski-Štovíček Correspondence for the Recollements of Purity\nby Ivo Herzog (Ohio State University\, USA) as part of ONCAS Online Non
commutative Algebra Seminar\n\n\nAbstract\nThe theory of purity for module
s over a ring R has been studied using the covariant\nas well as the contr
avariant functor categories. In their work on quiver Grassmanians\,\nCrawl
ey-Boevey and Sauter introduced a third such category\, the projective quo
tient \nfunctor category. We will explain how these three functor categori
es are related to the three equivalent definitions of purity. Ironically\,
the projective quotient functor category is closest in spirit to Prüfer'
s original definition. \n\nEach of these functor categories may be regarde
d as the middle term of a recollement of abelian categories whose localiza
tion/colocalization is given by the category R-Mod of R-modules. We will d
escribe the basic theory of recollements of functor categories and indicat
e how it reveals the common features of the three functor categories. Each
of these functor categories is related to the other two by a triangle of
Positselski-Štovíček correspondences\, which allows a detailed analysis
of its homological properties. \n\nThis is joint work with Xianhui Fu.\n
LOCATION:https://researchseminars.org/talk/ONCAS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Trlifaj (Charles University\, Prague)
DTSTART:20211012T160000Z
DTEND:20211012T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/6
DESCRIPTION:Title: Cl
osure properties of $\\displaystyle \\lim_\\longrightarrow \\mathcal C$\nby Jan Trlifaj (Charles University\, Prague) as part of ONCAS Online No
ncommutative Algebra Seminar\n\n\nAbstract\nLet $\\mathcal C$ be a class o
f (right $R$-) modules closed under finite direct sums. If $\\mathcal C$ c
onsists of finitely presented modules\, then the class $\\displaystyle \\l
im_\\longrightarrow \\mathcal C$ of all direct limits of modules from $\\m
athcal C$ is well-known to enjoy a number of closure properties. Moreover\
, if $R \\in \\mathcal C$\, $\\mathcal C$ consists of FP$_2$-modules\, and
$\\mathcal C$ is closed under extensions and direct summands\, then $\\di
splaystyle \\lim_\\longrightarrow \\mathcal C$ can be described homologica
lly: $\\displaystyle \\lim_\\longrightarrow \\mathcal C$ is the double per
pendicular class of $\\mathcal C$ with respect to the Tor$_1^R$ bifunctor
[1]. \n\nThings change completely when $\\mathcal C$ is allowed to contain
infinitely generated modules: $\\displaystyle \\lim_\\longrightarrow \\ma
thcal C$ then need not even be closed under direct limits. After presentin
g some positive general results (and their constraints)\, we will concentr
ate on two particular cases: $\\mathcal C = add(M)$ and $\\mathcal C = Add
(M)$\, for an arbitrary module $M$. We will prove that if $S = \\End M$ an
d $\\mathcal F$ is the class of all flat right $S$-modules\, then $\\displ
aystyle \\lim_\\longrightarrow add(M) = \\{ F \\otimes _S M \\mid F \\in \
\mathcal F \\}$. For $\\displaystyle \\lim_\\longrightarrow Add(M)$\, we w
ill have a similar formula\, involving the contratensor product $\\odot _S
$ and direct limits of projective right $S$-contramodules (for $S$ endowed
with the finite topology). We will also show that for various classes of
modules $\\mathcal D$\, if $M \\in \\mathcal D$ then $\\displaystyle \\lim
_\\longrightarrow add(M) = \\displaystyle \\lim_\\longrightarrow Add(M)$.
However\, the equality remains open in general\, even for (infinitely gene
rated) projective modules. \n\nThe talk is based on my recent joint work w
ith Leonid Positselski [2].\n\n[1] L.Angeleri Hügel\, J. Trlifaj: Direct
limits of modules of finite projective dimension\, in Rings\, Modules\, Al
gebras\, and Abelian Groups\, LNPAM 236\, M.Dekker\, New York 2004\, 27-44
.\n\n[2] L.Positelski\, J.Trlifaj: Closure properties of $\\displaystyle \
\lim_\\longrightarrow \\mathcal C$\, preprint.\n
LOCATION:https://researchseminars.org/talk/ONCAS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:André Leroy (University of Artois)
DTSTART:20211019T160000Z
DTEND:20211019T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/7
DESCRIPTION:Title: Co
mmutatively closed rings and their graphs\nby André Leroy (University
of Artois) as part of ONCAS Online Noncommutative Algebra Seminar\n\n\nAb
stract\nA subset S of a ring R is commutatively closed if for any elements
a\, b in R\, the product ab is in S if and only if the product ba is in S
. This concept was introduced in a recent paper and intended to have anoth
er perspective on Dedekind finite\, reversible\, semicommutative\, ... rin
gs. A topology was attached to this concept and in the present work we att
ach a graph and are able to compute the diameter of this graph for semisim
ple algebras. This answers some questions left open. \n\nThis is joint wor
k with Mona Abdi.\n
LOCATION:https://researchseminars.org/talk/ONCAS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Reyes (University of California)
DTSTART:20211102T160000Z
DTEND:20211102T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/8
DESCRIPTION:Title: Du
al coalgebras as a quantized maximal spectrum\nby Manuel Reyes (Univer
sity of California) as part of ONCAS Online Noncommutative Algebra Seminar
\n\n\nAbstract\nIf an algebra A has “many” finite-dimensional represen
tations\, we argue that its Sweedler dual coalgebra is a reasonable functo
rial quantization of the maximal spectrum of A. Many such algebras arise a
s twisted tensor products of commutative algebras\, including Ore extensio
ns and smash products. This leads to the problem of understanding the dual
coalgebra of a twisted tensor product. We will discuss when the Sweedler
dual of a twisted tensor product can be computed as a cross product coalge
bra\, a result that is achieved using methods of topological algebra.\n
LOCATION:https://researchseminars.org/talk/ONCAS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Van C Nguyen (United States Naval Academy)
DTSTART:20211116T170000Z
DTEND:20211116T180000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/9
DESCRIPTION:Title: Tw
isting of graded quantum groups and solutions to the quantum Yang-Baxter e
quation\nby Van C Nguyen (United States Naval Academy) as part of ONCA
S Online Noncommutative Algebra Seminar\n\n\nAbstract\nLet $H$ be a Hopf a
lgebra over a field $k$ such that $H$ is $\\mathbb Z$-graded as an algebra
. In this talk\, we introduce the notion of a twisting pair for $H$ and sh
ow that the Zhang twist of $H$ by such a pair can be realized as a 2-cocyc
le twist. We use twisting pairs to describe twists of Manin's universal qu
antum groups associated to quadratic algebras. Furthermore\, we discuss a
strategy to twist a solution to the quantum Yang-Baxter equation via the F
addeev-Reshetikhin-Takhtajan construction. If time permits\, we illustrate
this result for the quantized coordinate rings of $GL_n(k)$.\n
LOCATION:https://researchseminars.org/talk/ONCAS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Musson (University of Wisconsin Milwaukee)
DTSTART:20211130T170000Z
DTEND:20211130T180000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/10
DESCRIPTION:Title: H
ow to construct the lattice of submodules of a multiplicity free module fr
om partial information\nby Ian Musson (University of Wisconsin Milwauk
ee) as part of ONCAS Online Noncommutative Algebra Seminar\n\n\nAbstract\n
In general it is a difficult problem to construct the lattice of submodule
s $L(M )$ given module $M$. In [Sta12] a method is outlined for constuctin
g a distributive lattice from a knowledge of its join irreducibles. Howeve
r it is not an easy task to identify all join irreducible submodules of a
given module. In the case of a multiplicity free module M we present an al
ternative method based on the composition factors. As input we require a s
et of submodules $A_1\,\\ldots\, A_n$ whose submodule lattice is known\, w
hich contain all composition factors of $M$\, and for which all intersecti
ons $A_i \\cap A_j$ are known. From this we can reconstruct $L(M)$. We ill
ustrate the process for a Verma module M for the Lie superalgebra $\\mathf
rak{osp}(3\, 2)$. In this case\, $L(M)$ is isomorphic to the (extended) fr
ee distributive lattice of rank 3. This is well-known\, but quite complica
ted lattice. Indeed $M$\nhas 20 submodules and 8 composition factors\, eac
h with multiplicity one.\n\n$\\mathbf{Bibliography}$\n\n[Sta12] R. P. Stan
ley\, Enumerative combinatorics. Volume 1\, 2nd ed.\, Cambridge Studies in
Advanced Mathematics\, vol. 49\, Cambridge University Press\, Cambridge\,
2012. MR2868112\n
LOCATION:https://researchseminars.org/talk/ONCAS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason Gaddis (Miami University\, Ohio)
DTSTART:20211214T170000Z
DTEND:20211214T180000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/11
DESCRIPTION:Title: R
eflexive hull discriminants and applications\nby Jason Gaddis (Miami U
niversity\, Ohio) as part of ONCAS Online Noncommutative Algebra Seminar\n
\n\nAbstract\nIn algebraic number theory\, the discriminant is an importan
t invariant of a Galois field extension. There is a notion of the discrimi
nant for noncommutative algebras that are finite modules over their center
s. This has been used to solve several challenging problems\, such as to c
lassify the automorphism groups of certain families of noncommutative alge
bras. But the discriminant is notoriously difficult to compute in large ra
nk. In this talk\, I will review some of the history behind the discrimina
nt invariant and introduce a new notion\, the reflexive hull discriminant.
This modification has a geometric interpretation and\, moreover\, is well
-suited for algebras that are finitely generated but not necessarily free
over their centers. As an application\, I will show how this invariant can
be used to determine the automorphism groups for certain quantum generali
zed Weyl algebras.\n
LOCATION:https://researchseminars.org/talk/ONCAS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Posur (Munster University\, Germany)
DTSTART:20220215T170000Z
DTEND:20220215T180000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/13
DESCRIPTION:Title: O
n free abelian categories for theorem proving\nby Sebastian Posur (Mun
ster University\, Germany) as part of ONCAS Online Noncommutative Algebra
Seminar\n\n\nAbstract\nComputing explicitly within a free mathematical obj
ect can be interpreted as theorem proving. In this talk\, we discuss the c
onstructiveness of free abelian categories. A very concrete description of
free abelian categories was given by Murray Adelman\, and we demonstrate
how his description can be employed to validate homological lemmata like t
he Snake lemma computationally.\n
LOCATION:https://researchseminars.org/talk/ONCAS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Miodrag Iovanov (University of Iowa\, USA)
DTSTART:20220308T170000Z
DTEND:20220308T180000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/14
DESCRIPTION:Title: Q
uantum groups of finite representation type\nby Miodrag Iovanov (Unive
rsity of Iowa\, USA) as part of ONCAS Online Noncommutative Algebra Semina
r\n\n\nAbstract\nAlgebras of finite representation type - that is\, those
who have only finitely many indecomposable finite dimensional representati
ons up to isomorphism - have been of central interest in representation th
eory. Classically\, they appeared from work in modular representations\; a
(finite) group has finite representation type iff its p-Sylow subgroup is
cyclic. On the quantum side\, results of Farnsteiner describe the structu
re of finite group schemes (finite dimensional co-commutative Hopf algebra
s). Among the first examples of non-commutative and non-cocommutative quan
tum groups (Hopf algebras) are the Sweedler algebra and Taft algebras. The
se are pointed (their simple modules are 1-dimensional - they are points)\
, and they are also of finite type.\n\nTo study this in the generality of
infinite dimensional quantum groups (which includes gl_n\, quantum sl\, et
c.)\, one defines an algebra to be of finite type if given any dimension v
ector\, there are only finitely many indecomposables of this dimension vec
tor\; by the well known Brower-Thrall problems\, this is equivalent to the
above for finite dimensional algebras. We give an overview of various exa
mples of infinte quantum groups of finite type\, and give a complete class
ification of the pointed quantum groups of finite representation type. We
re-obtain results known for the finite dimensional case (including Taft al
gebras and their generalizations)\, and show that these include several in
teresting Hopf algebras\, such as those whose categories of comodules form
the category of chain complexes or the category of double chain complexes
\, and in general\, the list includes these and certain kind of twists and
deformations of theirs.\n
LOCATION:https://researchseminars.org/talk/ONCAS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pace Nielsen (Brigham Young University\, USA)
DTSTART:20220315T160000Z
DTEND:20220315T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/15
DESCRIPTION:Title: N
ilpotent polynomials with non-nilpotent coefficients\nby Pace Nielsen
(Brigham Young University\, USA) as part of ONCAS Online Noncommutative Al
gebra Seminar\n\n\nAbstract\nIt is well known that the coefficients of nil
potent polynomials over noncommutative rings generally are not all nilpote
nt. We show that this remains true even under extremely strong restrictio
ns on the set of nilpotents in the coefficient ring. If $R$ is a ring and
its set of nilpotents\, ${\\rm Nil}(R)$\, satisfies ${\\rm Nil}(R)^2=0$\,
then in general ${\\rm Nil}(R[x])\\not \\subseteq {\\rm Nil}(R)[x]$. Thi
s is proven by constructing an explicit polynomial example. The smallest
possible degree of such a polynomial is seven. Related problems are raise
d\, as well as connections to Kothe's conjecture and work of Agata Smoktun
owicz.\n
LOCATION:https://researchseminars.org/talk/ONCAS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shahn Majid (Queen Mary University of London\, UK)
DTSTART:20220329T160000Z
DTEND:20220329T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/16
DESCRIPTION:Title: Q
uantum jet bundles\nby Shahn Majid (Queen Mary University of London\,
UK) as part of ONCAS Online Noncommutative Algebra Seminar\n\n\nAbstract\n
We formulate a notion of jet bundles over a possibly noncommutative algebr
a \nA equipped with a torsion free connection. Among the conditions needed
for 3rd-order jets and above is that the connection also be flat and its
`generalised braiding tensor' σ:Ω1⊗AΩ1→Ω1⊗AΩ1 obey the Yang-Bax
ter equation or braid relations. We also cover the case of jet bundles of
a given `vector bundle' over A in the form of a bimodule E with flat bimod
ule connection with its braiding σEobeying the coloured braid relations.
Examples include the permutation group S3 with its 2-cycles calculus\, M2(
ℂ) and the bicrossproduct model quantum spacetime in two dimensions.\n
LOCATION:https://researchseminars.org/talk/ONCAS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henrik Holm (University of Copenhagen\, Denmark)
DTSTART:20220426T160000Z
DTEND:20220426T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/17
DESCRIPTION:Title: T
he Q-shaped derived category of a ring\nby Henrik Holm (University of
Copenhagen\, Denmark) as part of ONCAS Online Noncommutative Algebra Semin
ar\n\n\nAbstract\nThe derived category D(A) of the category Mod(A) of modu
les over a ring A is an important example of a triangulated category in al
gebra. It can be obtained as the homotopy category of the category Ch(A) o
f chain complexes of A-modules equipped with its standard model structure.
One can view Ch(A) as the category Fun(Q\,Mod(A)) of additive functors fr
om a certain small preadditive category Q to Mod(A). The model structure o
n Ch(A) = Fun(Q\,Mod(A)) is not inherited from a model structure on Mod(A)
but arises instead from the "self-injectivity" of the special category Q.
We will show that the functor category Fun(Q\,Mod(A)) has two interesting
model structures for many other self-injective small preadditive categori
es Q. These two model structures have the same weak equivalences\, and the
associated homotopy category is what we call the Q-shaped derived categor
y of A. We will also show that it is possible to generalize the homology f
unctors on Ch(A) to homology functors on Fun(Q\,Mod(A)) for most self-inje
ctive small preadditive categories Q. The talk is based on a joint paper w
ith Peter Jørgensen (arXiv:2101.06176)\, which has the same title as the
talk.\n
LOCATION:https://researchseminars.org/talk/ONCAS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Rothmaler (CUNY)
DTSTART:20220510T160000Z
DTEND:20220510T170000Z
DTSTAMP:20260422T225725Z
UID:ONCAS/18
DESCRIPTION:Title: E
very module has an Ulm length\nby Philipp Rothmaler (CUNY) as part of
ONCAS Online Noncommutative Algebra Seminar\n\n\nAbstract\nTo make sense o
f the statement in the title\, I introduce a concept of Ulm submodule that
grew out of discussions with Alex Martsinkovsky and applies to any module
over any associative ring with 1. In abelian groups it coincides with the
classical notion\, and so does the rest of the investigation. \n\nAs usua
l\, a module is said to have Ulm length 0 if it coincides with its own Ulm
submodule. These modules form a definable subcategory\, which\, over doma
ins\, coincides with that of divisible modules. The subcategory is equal t
o the entire category if and only if the ring is absolutely pure (on the s
ame side). Over RD-domains\, like Prüfer domains or the first Weyl algebr
a over a field of characteristic 0\, Ulm length 0 modules are injective.\n
\nTaking the Ulm submodule constitutes a functor\, which\, by iteration\,
leads to higher Ulm functors as usual and in turn to Ulm sequences and the
notion of Ulm length for any module. One of the main results is that the
(first) Ulm submodule of a pure-injective has length 0. In other words\,
pure-injectives have Ulm length at most 1\, just as over the integers. As
a consequence\, every module is a pure (even elementary) submodule of a mo
dule of Ulm length at most 1.\n\nAs another consequence one obtains\, for
any pure-injective over an RD domain\, a direct decomposition into a large
st injective submodule (= the first Ulm submodule)\, and a reduced module\
, that is\, a module with zero Ulm submodule.\n
LOCATION:https://researchseminars.org/talk/ONCAS/18/
END:VEVENT
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