Deformed Double Current Algebras via Deligne Categories

TL;DR

This paper introduces a new construction of Deformed Double Current Algebras using Deligne categories, providing an alternative perspective to the original definitions by Guay through ultraproducts of Cherednik algebra subalgebras.

Contribution

It offers a novel approach to constructing Deformed Double Current Algebras via Deligne categories and ultraproducts, expanding the understanding of their algebraic structure.

Findings

Constructed Deformed Double Current Algebras as endomorphism algebras in Deligne categories.

Provided an alternative to Guay's original algebraic definitions.

Connected the algebras to ultraproducts of Cherednik algebra subalgebras.

Abstract

In this paper we give an alternative construction of a certain class of Deformed Double Current Algebras. These algebras are deformations of $U({\rm End}(\Bbbk^r)[x,y])$ and they were initially defined and studied by N.Guay in his papers. Here we construct them as algebras of endomorphisms in Deligne category. We do this by taking an ultraproduct of spherical subalgebras of the extended Cherednik algebras of finite rank.