Difference operators of Sklyanin and van Diejen type

TL;DR

This paper explores the structure of difference operators related to the Sklyanin algebra, characterizing their coefficients, relations, and automorphisms, and establishing connections to van Diejen operators and finite-dimensional representations.

Contribution

It provides explicit conditions for difference operators to belong to Sklyanin representations, links these to van Diejen operators, and introduces new finite-dimensional representations.

Findings

Coefficients of Sklyanin difference operators have simple poles with linear residue relations.

Sum of Sklyanin generator products yields van Diejen type operators.

Kernel identities induce automorphisms and relate to finite-dimensional representations.

Abstract

The Sklyanin algebra $S_{\eta}$ has a well-known family of infinite-dimensional representations $D(\mu)$, $\mu \in C^*$, in terms of difference operators with shift $\eta$ acting on even meromorphic functions. We show that for generic $\eta$ the coefficients of these operators have solely simple poles, with linear residue relations depending on their locations. More generally, we obtain explicit necessary and sufficient conditions on a difference operator for it to belong to $D(\mu)$. By definition, the even part of $D(\mu)$ is generated by twofold products of the Sklyanin generators. We prove that any sum of the latter products yields a difference operator of van Diejen type. We also obtain kernel identities for the Sklyanin generators. They give rise to order-reversing involutive automorphisms of $D(\mu)$, and are shown to entail previously known kernel identities for the van Diejen operators. Moreover, for special $\mu$ they yield novel finite-dimensional representations of $S_{\eta}$.