Ordered Tensor Categories and Representations of the Mackey Lie Algebra of Infinite Matrices

We introduce (partially) ordered Grothendieck categories and apply results on their structure to the study of categories of representations of the Mackey Lie algebra of infinite matrices 𝔤𝔩^M(V, V_∗). Here 𝔤𝔩^M(V, V_∗) is the Lie algebra of endomorphisms of a nondegenerate pairing of countably infinite-dimensional vector spaces V_∗⊗ V→ K, where K is the base field. Tensor representations of 𝔤𝔩^M(V, V_∗) are defined as arbitrary subquotients of finite direct sums of tensor products (V^∗)^⊗m ⊗ (V_∗)^⊗n ⊗ V^⊗p where V^∗ denotes the algebraic dual of V. The category T𝔤𝔩M(V,V∗)3 which they comprise, extends a category T𝔤𝔩M(V,V∗) previously studied in Dan-Cohen et al. Adv. Math. 289, 205–278, (2016), Penkov and Serganova (2014) and Sam and Snowden Forum Math. Sigma 3(e11):108, (2015). Our main result is that T𝔤𝔩M(V,V∗)3 is a finite-length, Koszul self-dual, tensor category with a certain universal property that makes it into a “categorified algebra” defined by means of a handful of generators and relations. This result uses essentially the general properties of ordered Grothendieck categories, which yield also simpler proofs of some facts about the category T𝔤𝔩M(V,V∗) established in Penkov and Serganova (2014). Finally, we discuss the extension of T𝔤𝔩M(V,V∗)3 obtained by adjoining the algebraic dual (V_∗)^∗ of V_∗.