From Reflection Equation Algebra to Braided Yangians

TL;DR

This paper explores the properties of Reflection Equation algebras and braided Yangians, establishing Cayley-Hamilton identities and connecting to Lie algebras relevant for integrable models like the Gaudin system.

Contribution

It introduces a specific Cayley-Hamilton identity for Reflection Equation algebras and braided Yangians, linking these structures to classical Lie algebras and integrable models.

Findings

Derived a Cayley-Hamilton identity for the Reflection Equation algebra.

Established an analog of the Cayley-Hamilton identity for braided Yangians.

Constructed a Bethe subalgebra in the enveloping algebra related to the Gaudin model.

Abstract

In general, quantum matrix algebras are associated with a couple of compatible braidings. A particular example of such an algebra is the so-called Reflection Equation algebra. In this paper we analyse its specific properties, which distinguish it from other quantum matrix algebras (in first turn, from the RTT one). Thus, we exhibit a specific form of the Cayley-Hamilton identity for its generating matrix, which in a limit turns into the Cayley-Hamilton identity for the generating matrix of the enveloping algebra U(gl(m)). Also, we consider some specific properties of the braided Yangians, recently introduced by the authors. In particular, we establish an analog of the Cayley-Hamilton identity for the generating matrix of such a braided Yangian. Besides, by passing to a limit of the braided Yangian, we get a Lie algebra similar to that entering the construction of the rational Gaudin model. In its enveloping algebra we construct a Bethe subalgebra by the method due to D.Talalaev.