Universal tensor categories generated by dual pairs

Abstract

Let $V_{\ast }\otimes V\to \mathbb{C}$ be a non-degenerate pairing of countable-dimensional complex vector spaces $V$ and $V_{\ast }$. The Mackey Lie algebra $\mathfrak{g}={\mathfrak{gl}}^M(V,V_{\ast })$ corresponding to this paring consists of all endomorphisms $\phi$ of $V$ for which the space $V_{\ast }$ is stable under the dual endomorphism ${\phi }^{\ast }:V^{\ast }\to V^{\ast }$. We study the tensor Grothendieck category $\mathbb{T}$ generated by the $\mathfrak{g}$-modules $V$, $V_{\ast }$ and their algebraic duals $V^{\ast }$ and $V_{\ast }^{\ast }$. This 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 $X^{\prime }\hookrightarrow X$, $Y^{\prime }\hookrightarrow Y$ and a pairing $X\otimes Y\to \text{1}$ where \textbf{1} is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $\mathbb{T}$ and ${}_I\mathbb{T}$.

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