547 research outputs found
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 of finite non-degenerate
set-theoretic solutions to the Yang-Baxter equation. Namely, we
construct a finite quotient of , generalizing
the Coxeter-like groups introduced by Dehornoy for involutive solutions. This
yields a finitary setting for testing injectivity: if injects into
, then it also injects into . We shrink every
solution to an injective one with the same structure group, and compute the
rank of the abelianization of . 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
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