Representation categories of Mackey Lie algebras as universal monoidal categories

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 $A$ is equipped with a possibly transfinite filtration together with a pairing $A \otimes B \to \mathbb{1}$. 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 $t \in \mathbb{Z}_{\geq 0}$.

Keywords

Mackey Lie algebra, tensor module, monoidal category, Koszul algebra, semi-artinian, Grothendieck category

2010 Mathematics Subject Classification

16S37, 16T15, 17B10, 17B65, 18D10, 18E15

Full Text (PDF format)

Received 3 October 2017

Published 14 September 2018