Representation categories of Mackey Lie algebras as universal monoidal categories

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. We study a monoidal category ${\mathbb{T}}_{\alpha }$ which is universal among all symmetric $\mathbb{K}$-linear monoidal categories generated by two objects $A$ and $B$ such that $A$ has a, possibly transfinite, filtration. We construct ${\mathbb{T}}_{\alpha }$ as a category of representations of the Lie algebra ${\mathfrak{gl}}^M(V_{\ast },V)$ consisting of endomorphisms of a fixed diagonalizable pairing $V_{\ast }\otimes V\to \mathbb{K}$ of vector spaces $V_{\ast }$ and $V$ of dimension $\alpha$. Here $\alpha$ is an arbitrary cardinal number. We describe explicitly the simple and the injective objects of ${\mathbb{T}}_{\alpha }$ and prove that the category ${\mathbb{T}}_{\alpha }$ is Koszul. We pay special attention to the case where the filtration on $A$ is finite. In this case $\alpha ={\aleph }_t$ for $t\in {\mathbb{Z}}_{\ge 0}$.

arXiv comment32 pages + references