Intertwining operators for Sklyanin algebra and elliptic hypergeometric series

TL;DR

This paper constructs intertwining operators for the Sklyanin algebra using elliptic hypergeometric series, which form the basis for an elliptic R-matrix satisfying the Yang-Baxter equation, advancing integrable systems theory.

Contribution

It introduces explicit elliptic hypergeometric series-based intertwining operators for infinite-dimensional Sklyanin algebra representations, linking them to the elliptic R-matrix and Yang-Baxter equation.

Findings

Constructed W-operators using elliptic hypergeometric series.

Derived an elliptic R-matrix satisfying Yang-Baxter equation.

Provided a graphical representation of the algebraic structures.

Abstract

Intertwining operators for infinite-dimensional representations of the Sklyanin algebra with spins l and -l-1 are constructed using the technique of intertwining vectors for elliptic L-operator. They are expressed in terms of elliptic hypergeometric series with operator argument. The intertwining operators obtained (W-operators) serve as building blocks for the elliptic R-matrix which intertwines tensor product of two L-operators taken in infinite-dimensional representations of the Sklyanin algebra with arbitrary spin. The Yang-Baxter equation for this R-matrix follows from simpler equations of the star-triangle type for the W-operators. A natural graphic representation of the objects and equations involved in the construction is used.