Olshanski's Centralizer Construction and Deligne Tensor Categories

Abstract

The family of Deligne tensor categories $\mathrm{Rep}(G{L}_t)$ 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 $G{L}_t$-invariants in the universal enveloping algebra $U(\mathfrak{g}{\mathfrak{l}}_{\mathfrak{t}+\mathfrak{n}})$ is the tensor product of $Y({\mathfrak{gl}}_n)$ and the center of $U(\mathfrak{g}{\mathfrak{l}}_{\mathfrak{t}})$. The main feature of this construction is that it does not involve passing to a limit, contrary to the original construction of Olshanski.

11 pages