Fomin-Kirillov algebras

This is an extended abstract of the talk given in the Oberwolfach miniworkshop "Nichols algebras and Weyl groupoids" in October 2012.Comment: 2 page

Nichols algebras over groups with finite root system of rank two II

We classify all non-abelian groups G for which there exists a pair (V,W) of absolutely simple Yetter–Drinfeld modules over G such that the Nichols algebra of the direct sum of V and W is finite-dimensional, under two assumptions: the square of the braiding between V and W is not the identity, and G is generated by the support of V and W. As a corollary, we prove that the dimensions of such V and W are at most six. As a tool we use the Weyl groupoid of (V,W).Fil: Heckenberger, István. Philipps Universität Marburg; AlemaniaFil: Vendramin, Claudio Leandro. Philipps Universität Marburg; Alemania. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Skew braces: a brief survey

Our primary focus is on the theory of skew braces, specifically exploring their connection with combinatorial solutions to the Yang-Baxter equation. Skew braces have recently emerged as intriguing algebraic structures, and their link to the Yang-Baxter equation adds further depth and significance to their study. Throughout this article, we place particular emphasis on various problems and conjectures that arise within this field. These open questions serve to stimulate further research and investigation, as we strive to enhance our understanding of skew braces and their role in the study of the Yang-Baxter equation.Comment: 22 pages.Typos corrected. XL Workshop on Geometric Methods in Physics, Bia{\l}owie\.za 2023. Postprin

Isoclinism of skew braces

We define isoclinism of skew braces and present several applications. We study some properties of skew braces that are invariant under isoclinism. For example, we prove that right nilpotency is an isoclinism invariant. This result has application in the theory of set-theoretic solutions to the Yang-Baxter equation. We define isoclinic solutions and study multipermutation solutions under isoclinism.Comment: 14 pages. Postprint versio

PBW deformations of a Fomin–Kirillov Algebra and other examples

We begin the study of PBW deformations of graded algebras relevant to the theory of Hopf algebras. One of our examples is the Fomin–Kirillov algebra E3. Another one appeared in a paper of García Iglesias and Vay. As a consequence of our methods, we determine when the deformations are semisimple and we are able to produce PBW bases and polynomial identities for these deformations.Fil: Heckenberger, I.. Philipps-Universität Marburg; AlemaniaFil: Vendramin, Claudio Leandro. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Finite-dimensional Nichols algebras of simple Yetter-Drinfeld modules (over groups) of prime dimension

Acknowledgements. EM would like to thank Ben Martin for fruitful discussions about geometric invariant theory in positive characteristicPeer reviewe

Combinatorial solutions to the reflection equation

We use ring-theoretic methods and methods from the theory of skew braces to produce set-theoretic solutions to the reflection equation. We also use set-theoretic solutions to construct solutions to the parameter-dependent reflection equation.Comment: 20 pages. Final versio

A characterization of finite multipermutation solutions of the Yang-Baxter equation

We prove that a finite non-degenerate involutive set-theoretic solution (X, r) of the Yang{Baxter equation is a multipermutation solution if and only if its structure group G(X, r) admits a left ordering or equivalently it is poly-Z.Fil: Bachiller, David. Universitat Autònoma de Barcelona; EspañaFil: Cedó, Ferran. Universitat Autònoma de Barcelona; EspañaFil: Vendramin, Claudio Leandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin