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SUMMARY:Ramón Esteban-Romero (Polytechnic University of Valencia)
DTSTART:20201119T160000Z
DTEND:20201119T170000Z
DTSTAMP:20260423T053015Z
UID:GOThIC/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GOThIC/7/">T
 riply factorised groups and skew left braces</a>\nby Ramón Esteban-Romero
  (Polytechnic University of Valencia) as part of GOThIC - Ischia Online Gr
 oup Theory Conference\n\n\nAbstract\nThe Yang-Baxter equation is a consist
 ency equation of the statistical mechanics\nproposed by Yang [Yang67] and 
 Baxter\n[Baxter73] that describes the interaction of many particles in som
 e scattering\nsituations. This equation lays the foundation for the theory
  of quantum\ngroups and Hopf algebras. During the last years\, the study s
 uggested by\nDrinfeld [Drinfeld92] of the so-called\nset-theoretic solutio
 ns of the Yang-Baxter equation has motivated the\nappearance of many algeb
 raic structures. Among these structures we\nfind the *skew left braces*\, 
 introduced by Guarnieri and\nVendramin [GuarnieriVendramin17] as a general
 isation of the\nstructure  of left brace defined by Rump [Rump07]. It cons
 ists of a set $B$\nwith two operations $+$ and $\\cdot$\, not necessarily 
 commutative\, that give $B$ two structures of\ngroup linked by a modified 
 distributive law.\n\nThe multiplicative group $C=(B\, {\\cdot})$ of a skew
  left brace $(B\, {+}\, {\\cdot})$\nacts on the\nmultiplicative group $K=(
 B\, {+})$ by means of an action $\\lambda\\colon\nC\\longrightarrow \\oper
 atorname{Aut}(K)$ given by\n$\\lambda(a)(b)=-a+a\\cdot b$\, for $a$\, $b\\
 in B$. With respect to this\naction\, the identity map $\\delta\\colon C\\
 longrightarrow K$ becomes a\nderivation or $1$-cocycle with respect to $\\
 lambda$. In the semidirect\nproduct $G=[K]C=\\{(k\,c)\\mid k\\in K\, c\\in
  C\\}$\, there is a\ndiagonal-type subgroup $D=\\{(\\delta(c)\, c)\\mid c\
 \in C\\}$ such that\n$G=KD=CD$\, $K\\cap D=C\\cap D=1$. This approach was 
 presented by\nSysak in [Sysak11-PortoCesareo] and motivates the use of\nte
 chniques of group theory to study skew left braces.\n\nWe present in this 
 talk some applications of this approach to obtain\nsome results about skew
  left braces. These results have been obtained\nin collaboration with Adol
 fo Ballester-Bolinches.\n\nThis work has been supported by the research gr
 ants\nPGC2018-095140-B-I00 from the Ministerio de Ciencia\,\n  Innovaci\\'
 on y Universidades (Spanish Government)\, the\nAgencia Estatal de Investig
 aci\\'on (Spain)\, and FEDER (European\nUnion)\, and PROMETEO/2017/057 fro
 m the Generalitat\n(Valencian Community\, Spain).\n\nReferences\n\n[Baxter
 73] R. Baxter. Eight-vertex model in lattice statistics and one-dimensiona
 l\nanisotropic Heisenberg chain. I. Some fundamental eigenvectors. Ann.\nP
 hysics\, 76(1):1–24\, 1973.\n\n[Drinfeld92] V. G. Drinfeld. On some unso
 lved problems in quantum group theory.\nIn P. P. Kulish\, editor\, Quantum
  groups. Proceedings of workshops held\nin the Euler International Mathema
 tical Institute\, Leningrad\, fall 1990\,\nvolume 1510 of Lecture Notes in
  Mathematics\, pages 1–8. Springer-Verlag\,\nBerlin\, 1992.\n\n[Guarnier
 iVendramin17] L. Guarnieri and L. Vendramin. Skew-braces and the Yang-Baxt
 er equation. Math. Comp.\, 86(307):2519–2534\, 2017.\n\n[Rump07] W. Rump
 . Braces\, radical rings\, and the quantum Yang-Baxter equation.\nJ. Algeb
 ra\, 307:153–170\, 2007.\n\n[Sysak11-PortoCesareo] Y. P. Sysak. Products
  of groups and quantum Yang-Baxter equation.\nNotes of a talk in Advances 
 in Group Theory and Applications\, Porto\nCesareo\, Lecce\, Italy\, 2011.\
 n\n[Yang67] C. N. Yang. Some exact results for many-body problem in one di
 mension\nwith repulsive delta-function interaction. Phys. Rev. Lett\, 19:1
 312–1315\,\n1967\n
LOCATION:https://researchseminars.org/talk/GOThIC/7/
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