Universal Tensor Categories Generated by Dual Pairs

Let V_∗⊗ V→ C be a non-degenerate pairing of countable-dimensional complex vector spaces V and V_∗. The Mackey Lie algebra g= gl^M(V, V_∗) corresponding to this pairing consists of all endomorphisms φ of V for which the space V_∗ is stable under the dual endomorphism φ^∗: V^∗→ V^∗. We study the tensor Grothendieck category T generated by the g-modules V, V_∗ and their algebraic duals V^∗ and V∗∗. The category T is an analogue of categories considered in prior literature, the main difference being that the trivial module C is no longer injective in T. We describe the injective hull I of C in T, and show that the category T is Koszul. In addition, we prove that I is endowed with a natural structure of commutative algebra. We then define another category _IT of objects in T which are free as I-modules. Our main result is that the category _IT is also Koszul, and moreover that _IT is universal among abelian C-linear tensor categories generated by two objects X, Y with fixed subobjects X^′↪ X, Y^′↪ Y and a pairing X⊗ Y→ 1 where 1 is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories T and _IT.