# Olshanski's Centralizer Construction and Deligne Tensor Categories

## TL;DR

This paper generalizes Olshanski's centralizer construction of the Yangian to Deligne tensor categories, establishing a new algebraic relationship without taking limits, thus broadening the scope of classical representation theory results.

## Contribution

It introduces a limit-free generalization of Olshanski's centralizer construction within Deligne tensor categories, connecting Yangians and universal enveloping algebras for complex parameters.

## Key findings

- Centralizer subalgebra of $GL_t$-invariants is isomorphic to a tensor product of $Y(rak{gl}_n)$ and the center of $U(rak{gl}_t)$.
- The construction applies for generic complex $t$, avoiding the limit process used in classical approaches.
- Provides a new algebraic framework extending classical representation theory to complex rank categories.

## Abstract

The family of Deligne tensor categories $\mathrm{Rep}(GL_t)$ is obtained from the categories $\mathbf{Rep}~GL(n)$ of finite dimensional representations of groups $GL(n)$ by interpolating the integer parameter $n$ to complex values. Therefore, it is a valuable tool for generalizing classical statements of representation theory. In this work we introduce and prove the generalization of Olshanski's centralizer construction of the Yangian $Y(\mathfrak{gl}_n)$. Namely, we prove that for generic $t\in\mathbb{C}$ the centralizer subalgebra of $GL_t$-invariants in the universal enveloping algebra $U(\mathfrak{gl_{t+n}})$ is the tensor product of $Y(\mathfrak{gl}_n)$ and the center of $U(\mathfrak{gl_{t}})$. The main feature of this construction is that it does not involve passing to a limit, contrary to the original construction of Olshanski.