Yangian Algebras and Classical Riemann Problems

TL;DR

This paper explores various Hopf algebras linked to Yang's quantum Yang-Baxter solution, focusing on Riemann problems for currents to define and differentiate algebra structures and their representations.

Contribution

It introduces two Riemann problems leading to new algebraic structures and analyzes their properties and differences in representation theory.

Findings

Defined central extended Yangian double for sl_2

Identified degeneration of elliptic affine algebra

Compared properties of different algebraic structures

Abstract

We investigate different Hopf algebras associated to Yang's solution of quantum Yang-Baxter equation. It is shown that for the precise definition of the algebra one needs the commutation relations for the deformed algebra of formal currents and the specialization of the Riemann problem for the currents. Two different Riemann problems are considered. They lead to the central extended Yangian double associated with ${sl}_2$ and to the degeneration of scaling limit of elliptic affine algebra. Unless the defining relations for the generating functions of the both algebras coincide their properties and the theory of infinite-dimensional representations are quite different. We discuss also the Riemann problem for twisted algebras and for scaled elliptic algebra.