In around 2005, Wolfgang Rump introduced braces, a generalization of nilpotent rings to describe all involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation. This formulation then rapidly found application in other research areas. This talk will review these applications.
Definition. A set A with binary operations of addition +, and multiplication ∘ is a brace if (A, +) is an abelian group, (A, ∘) is a group and a ∘ (b+c)+a = a ∘ b+a ∘ c for every a, b, c ∈ A. It follows from this definition that every nilpotent ring with the usual addition and with multiplication a ∘ b = ab + a + b is a brace.
Braces have been shown to be equivalent to several concepts in group theory such as groups with bijective 1-cocycles and regular subgroups of the holomorph of abelian groups. In algebraic number theory there is a correspondence between braces and Hopf-Galois extensions of abelian type first observed by David Bachiller. There is also connection between R-braces and pre-Lie algebras discovered by Wolfgang Rump in 2014. One generator braces have been shown to describe indecomposable, involutive solutions of the Yang-Baxter equation.
On the other hand, Anastasia Doikou and Robert Weston have recently discovered some fascinating connections between braces and quantum integrable systems. In particular, to find solutions of the set-theoretic reflection equation it is needed to solve problems on some polynomial identities in nilpotent rings. Because previously the theory of polynomial identities was mainly developed for prime rings, and for the reflection equation we only consider nilpotent rings, there are no known methods for solving such problems. We will mention some open problems on polynomial identities in nilpotent rings which appear in this situation.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
