Topological tensor representations of gl(V) for a space V of countable dimension

TL;DR

This paper develops a tensor category framework for topological representations of the infinite-dimensional Lie algebra gl(V), establishing an equivalence with tensor representations of finitary infinite matrices, thus extending finite-dimensional Lie theory.

Contribution

It introduces a new tensor category for topological representations of gl(V) and proves its equivalence to representations of infinite matrices, generalizing finite-dimensional Lie algebra modules.

Findings

Established a tensor category for topological gl(V) representations.

Proved the category is equivalent to representations of infinite matrices.

Extended finite-dimensional Lie algebra representation theory to infinite dimensions.

Abstract

The Lie algebra $gl(V)$ is the Lie algebra of all endomorphisms of a countable-dimensional complex vector space $V$. We define a tensor category of topological representations of the Lie algebra $gl(V)$, so that $V$, its dual and the adjoint representation $gl(V)$ are objects of this category. This makes it an analogue of the category of finite-dimensional modules over the finite-dimensional Lie algebra $gl(n)$. Our main result is that this category is antiequivalent as a symmetric monoidal category to the category of tensor representations of the Lie algebra of finitary infinite matrices.