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

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Summary

Introduction

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

Notation
Plethysm
Ordered Grothendieck Categories
Tensor Categories
Simple Objects and Their Endomorphism Algebras
The Category T
Injective Resolutions
Koszulity
An Internal Commutative Algebra and Its Modules
The Category IT
Universality
Orthogonal and Symplectic Analogues of the Categories T and IT
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