On deformed double current algebras for simple Lie algebras

TL;DR

This paper establishes the equivalence of two different presentations of deformed double current algebras for simple Lie algebras, constructs a central element, and explores the large center in type A for specific parameters.

Contribution

It proves the equivalence of two presentations of deformed double current algebras and identifies a significant central element, advancing understanding of their structure.

Findings

Proved the equivalence of two algebra presentations.

Constructed a specific central element.

Identified a large center in type A for certain parameters.

Abstract

We prove the equivalence of two presentations of deformed double current algebras associated to a complex simple Lie algebra, the first one obtained via a degeneration of affine Yangians while the other one naturally appeared in the construction of the elliptic Casimir connection. We also construct a specific central element of these algebras and, in type A, show that they contain a very large center for certain values of their parameters.