Set-theoretic solutions of the Yang-Baxter equation and new classes of R-matrices

We describe several methods of constructing R-matrices that are dependent upon many parameters, for example unitary R-matrices and R-matrices whose entries are functions. As an application, we construct examples of R-matrices with prescribed singular values. We characterise some classes of indecomposable set-theoretic solutions of the quantum Yang-Baxter equation (QYBE) and construct R-matrices related to such solutions. In particular, we establish a correspondence between one-generator braces and indecomposable, non-degenerate involutive set-theoretic solutions of the QYBE, showing that such solutions are abundant. We show that R-matrices related to involutive, non-degenerate solutions of the QYBE have special form. We also investigate some linear algebra questions related to R-matrices.Comment: Corrected typos and improved presentatio

More on skew braces and their ideals

This paper introduces the notion of a strongly prime ideal, and shows that thelargest solvable ideal in a finite brace equals the intersection of all strongly primeideals in this brace. This is used to generalise some well known results from ringtheory into the context of braces and pre-Lie algebras. Several open questions arealso posed

JACOBSON RADICAL ALGEBRAS WITH QUADRATIC GROWTH

AbstractWe show that over every countable algebraically closed field $\mathbb{K}$ there exists a finitely generated $\mathbb{K}$-algebra that is Jacobson radical, infinite-dimensional, generated by two elements, graded and has quadratic growth. We also propose a way of constructing examples of algebras with quadratic growth that satisfy special types of relations.</jats:p