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

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.