Lie algebras and Yang-Baxter equations

Abstract

At the previous congress (CRM 6), we reviewed the constructions of Yang-Baxter operators from associative algebras and presented some (colored) bialgebras and Yang-Baxter systems related to them. The current talk deals with Yang-Baxter operators from (G, θ)-Lie algebras (structures that unify the Lie algebras and Lie superalgebras). Thus, we produce solutions for the constant and the spectral-parameter Yang-Baxter equations, Yang-Baxter systems, etc. Attempting to present the general framework we review the work of other authors and we propose problems, applications, and directions of study.