Topological semiinfinite tensor (super)modules

TL;DR

This paper constructs and analyzes universal monoidal categories of topological tensor supermodules over Lie superalgebras related to Tate spaces, establishing various equivalences and universality properties.

Contribution

It introduces new monoidal categories for topological tensor supermodules over Lie superalgebras and proves their universality and equivalence properties, extending previous categorical frameworks.

Findings

Categories are anti-equivalent to previously studied ones.

Established equivalences between categories of different Lie superalgebras.

Proved universality properties of the constructed categories.

Abstract

We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras $\mathfrak{gl}(V\oplus \Pi V)$ and $\mathfrak{osp}(V\oplus \Pi V)$ associated with a Tate space $V$. Here $V\oplus \Pi V$ is a $\mathbb{Z}/2\mathbb{Z}$-graded topological vector space whose even and odd parts are isomorphic to $V$. We discuss the purely even case first, by introducing monoidal categories$\widehat{\mathbf{T}}_{\mathfrak{gl}(V)}$, $\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and $\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$, and show that these categories are anti-equivalent to respective previously studied categories $\mathbb{T}_{\mathfrak{gl}(V)}$, $\mathbb{T}_{\mathfrak{o}(V)}$, $\mathbb{T}_{\mathfrak{sp}(V)}$. These latter categories have certain universality properties as monoidal categories, which consequently carry over to $\widehat{\mathbf{T}}_{\mathfrak{gl}(V)}$, $\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and $\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$. Moreover, the categories $\mathbb{T}_{\mathfrak{o}(V)}$ and $\mathbb{T}_{\mathfrak{sp}(V)}$ are known to be equivalent, and this implies the equivalence of the categories $\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and $\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$. After introducing a supersymmetric setting, we establish the equivalence of the category $\widehat{\mathbf{T}}_{\mathfrak{gl}(V)}$ with the category $\widehat{\mathbf{T}}_{\mathfrak{gl}(V\oplus \Pi V)}$, and the equivalence of both categories $\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and $\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$ with $\widehat{\mathbf{T}}_{\mathfrak{osp}(V\oplus \Pi V)}$.