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

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 FK3. Another one appeared in a paper of Garc\'ia 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.Comment: 22 pages. Accepted for publication in Algebr. Represent. Theor

On structure groups of set-theoretic solutions to the Yang-Baxter equation

This paper explores the structure groups $G_{(X,r)}$ of finite non-degenerate set-theoretic solutions $(X,r)$ to the Yang-Baxter equation. Namely, we construct a finite quotient $\overline{G}_{(X,r)}$ of $G_{(X,r)}$, generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if $X$ injects into $G_{(X,r)}$, then it also injects into $\overline{G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of $G_{(X,r)}$. We show that multipermutation solutions are the only involutive solutions with diffuse structure group; that only free abelian structure groups are biorderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: biorderable, left-orderable, abelian, free abelian, torsion free.Comment: 32 pages. Final version. Accepted for publication in Proc. Edinburgh Math. So

Cohomology and extensions of braces

Braces and linear cycle sets are algebraic structures playing a major role in the classification of involutive set-theoretic solutions to the Yang-Baxter equation. This paper introduces two versions of their (co)homology theories. These theories mix the Harrison (co)homology for the abelian group structure and the (co)homology theory for general cycle sets, developed earlier by the authors. Different classes of brace extensions are completely classified in terms of second cohomology groups.Comment: 16 pages. Final version. Accepted for publication in Pacific Journal of Mathematic

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