Folded and contracted solutions of coupled classical dynamical Yang-Baxter and reflection equations

TL;DR

This paper presents a method to construct algebra-valued solutions to coupled classical dynamical Yang-Baxter and reflection equations, generalizing boundary KZB equations through folding and contracting symmetries.

Contribution

It introduces a concrete recipe for constructing solutions to coupled dynamical equations using folding and contracting techniques, extending the understanding of boundary integrable systems.

Findings

Constructed explicit triples satisfying coupled equations

Connected solutions to boundary KZB equations and Lie algebra representations

Determined specific dynamical r-matrices for simple Lie algebras

Abstract

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space $\mathfrak{a}$ satisfying three coupled classical dynamical Yang-Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on $\mathfrak{a}$, generalising asymptotic boundary Knizhnik-Zamolodchikov-Bernard (KZB) equations. The recipe involves folding and contracting $\mathfrak{a}$-invariant and $\theta$-twisted symmetric classical dynamical $r$-matrices along an involutive automorphism $\theta$. In case of the universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ we determine the Etingof-Schiffmann classical dynamical $r$-matrices which are $\mathfrak{a}$-invariant and $\theta$-twisted symmetric. The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.