The representation theory of Brauer categories II: curried algebra

TL;DR

This paper explores a new perspective on algebraic representations by currying, which simplifies the conditions and extends to tensor categories, revealing connections between combinatorial categories and Lie algebras.

Contribution

It introduces the concept of curried algebra representations in tensor categories, linking combinatorial categories like the Brauer category to Lie algebra structures.

Findings

Brauer category is the curried form of the symplectic Lie algebra

Currying provides a dual-free formulation of algebra representations

New applications and insights in tensor category theory

Abstract

A representation of $\mathfrak{gl}(V)=V \otimes V^*$ is a linear map $\mu \colon \mathfrak{gl}(V) \otimes M \to M$ satisfying a certain identity. By currying, giving a linear map $\mu$ is equivalent to giving a linear map $a \colon V \otimes M \to V \otimes M$, and one can translate the condition for $\mu$ to be a representation to a condition on $a$. This alternate formulation does not use the dual of $V$, and makes sense for any object $V$ in a tensor category $\mathcal{C}$. We call such objects representations of the curried general linear algebra on $V$. The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category "is" the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore.