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SUMMARY:Arne Van Antwerpen (Ghent University\, Belgium)
DTSTART:20240930T150000Z
DTEND:20240930T160000Z
DTSTAMP:20260423T035611Z
UID:ENAAS/91
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ENAAS/91/">I
 ndecomposable and simple solutions of the Yang-Baxter equation</a>\nby Arn
 e Van Antwerpen (Ghent University\, Belgium) as part of European Non-Assoc
 iative Algebra Seminar\n\n\nAbstract\nRecall that a set-theoretic solution
  of the Yang-Baxter equation is a tuple $(X\,r)$\, where $X$ is a non-empt
 y set and $r: X \\times X \\rightarrow X \\times X$ a bijective map such t
 hat $$(r \\times id_X ) (id_X \\times r) (r \\times id_X) = (id_X \\times 
 r) (r \\times id_X ) (id_X \\times r)\,$$ where one denotes $r(x\,y)=(\\la
 mbda_x(y)\, \\rho_y(x))$. Attention is often restricted to so-called non-d
 egenerate solutions\, i.e. $\\lambda_x$ and $\\rho_y$ are bijective. We wi
 ll call these solutions for short in the remainder of this abstract. To un
 derstand more general objects\, it is an important technique to study 'min
 imal' objects and glue these together. For solutions both indecomposable a
 nd simple solutions fit the bill for being a minimal object. In this talk 
 we will report on recent work with I. Colazzo\, E. Jespers and L. Kubat on
  simple solutions. In particular\, we will discuss an extension of a resul
 t of M. Castelli that allows to identify whether a solution is simple\, wi
 thout having to know or calculate all smaller solutions. This method emplo
 ys so-called skew braces\, which were constructed to provide more examples
  of solutions\, but also govern many properties of general solutions. In t
 he latter part of the talk\, we discuss the extension of a method to const
 ruct new indecomposable or simple solutions from old ones via cabling\, or
 iginally introduced by V. Lebed\, S. Ramirez and L. Vendramin to unify the
  known results on indecomposability of solutions.\n
LOCATION:https://researchseminars.org/talk/ENAAS/91/
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