Abstract
Let V_*otimes Vrightarrow {mathbb {C}} be a non-degenerate pairing of countable-dimensional complex vector spaces V and V_*. The Mackey Lie algebra {mathfrak {g}}=mathfrak {gl}^M(V,V_*) corresponding to this pairing consists of all endomorphisms varphi of V for which the space V_* is stable under the dual endomorphism varphi ^*: V^*rightarrow V^*. We study the tensor Grothendieck category {mathbb {T}} generated by the {mathfrak {g}}-modules V, V_* and their algebraic duals V^* and V^*_*. The category {{mathbb {T}}} is an analogue of categories considered in prior literature, the main difference being that the trivial module {mathbb {C}} is no longer injective in {mathbb {T}}. We describe the injective hull I of {mathbb {C}} in {mathbb {T}}, and show that the category {mathbb {T}} is Koszul. In addition, we prove that I is endowed with a natural structure of commutative algebra. We then define another category _I{mathbb {T}} of objects in {mathbb {T}} which are free as I-modules. Our main result is that the category {}_I{mathbb {T}} is also Koszul, and moreover that {}_I{mathbb {T}} is universal among abelian {mathbb {C}}-linear tensor categories generated by two objects X, Y with fixed subobjects X'hookrightarrow X, Y'hookrightarrow Y and a pairing Xotimes Yrightarrow {mathbf{1 }} where 1 is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories {mathbb {T}} and {}_I{mathbb {T}}.
Highlights
A tensor category for us is a symmetric, not necessarily rigid, C-linear monoidal abelian category
In [14] the observation was made that the four representations V, V∗, V ∗, V∗∗ generate a finite-length tensor category T4glM (V,V∗) over the larger Lie algebra glM (V, V∗), see Sect
A main difference of the category T4glM (V,V∗) with previously studied categories is that, as we show in the present paper, the injective hulls of simple objects are not objects of T4glM (V,V∗) but of a colimit-completion of T4glM (V,V∗) which we denote by T
Summary
A tensor category for us is a symmetric, not necessarily rigid, C-linear monoidal abelian category. 2. in [14] the observation was made that the four representations V , V∗, V ∗, V∗∗ generate a finite-length tensor category T4glM (V ,V∗) over the larger Lie algebra glM (V , V∗), see Sect. One can assume in addition that the pairing X ⊗ X → 1 is symmetric or antisymmetric, which leads to new universality problems for tensor categories With this in mind, we introduce T2o(V ) and T2sp(V ) where o(V ) and sp(V ) are respective orthogonal and symplectic Lie algebras of a countable-dimensional vector space V. The categories Io(V) T2 and Isp(V) T2 are canonically equivalent as monoidal categories
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