A set-theoretic solution of the Yang-Baxter equation is a mapping r : X×X → X×X satisfying (r×1)(1×r)(r×1) = (1×r)(r×1)(1×r). A solution r : (x, y) → (σx(y), τy (x)) is called non-degenerate if the mappings σx and τy are permutations, for all x, y ∈ X. A solution is called involutive if r2 = 1. If (X, r) is a non-degenerate involutive solution (X, r) then the relation ∼ defined by x ∼ y ≡ σx = σy is a congruence. A solution is of multipermutation level 2 if |(X/ ∼)/ ∼ | = 1. In our talk, we focus on these solutions and we present several constructions and properties.
This video is part of the European Non-Associative Algebra Seminar series.
