The representation theory of Brauer categories II: Curried algebra

A representation of gl(V)=V ⊗ V^{} is a linear map μ gl(V) ⊗ M → M satisfying a certain identity. By currying, giving a linear map μ is equivalent to giving a linear map a V ⊗ M → V ⊗ M , and one can translate the condition for μ to be a representation into 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 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.