First fundamental theorems of invariant theory for quantum supergroups

TL;DR

This paper establishes fundamental theorems of invariant theory for quantum supergroups related to rak{gl}_{m|n} and rak{osp}_{m|2n}, demonstrating the fullness of a tensor functor and spanning properties of hom-spaces.

Contribution

It proves the fullness of a tensor functor from ribbon graphs to quantum supergroup representations and describes spanning sets for hom-spaces in specific cases.

Findings

The tensor functor is full for rak{g}=rak{gl}_{m|n} or rak{osp}_{2\u03bb+1|2n}.

Hom-spaces are spanned by ribbon graph images when r+s<2rak{g} conditions.

The proofs use equivalence of module categories for different quantizations of rak{g}.

Abstract

Let $U_q(\mathfrak{g})$ be the quantum supergroup of $\mathfrak{gl}_{m|n}$ or the modified quantum supergroup of $osp_{m|2n}$ over the field of rational functions in $q$, and let $V_q$ be the natural module for $U_q(\mathfrak{g})$. There exists a unique tensor functor, associated with $V_q$, from the category of ribbon graphs to the category of finite dimensional representations of $U_q(\mathfrak{g}$, which preserves ribbon category structures. We show that this functor is full in the cases $\mathfrak{g}=\mathfrak{gl}_{m|n}$ or $osp_{2\ell+1|2n}$. For $\mathfrak{g}=osp_{2\ell|2n}$, we show that the space $Hom_{U_q(\mathfrak{g}}(V_q^{\otimes r}, V_q^{\otimes s})$ is spanned by images of ribbon graphs if $r+s< 2\ell(2n+1)$. The proofs involve an equivalence of module categories for two versions of the quantisation of $U(\mathfrak{g})$.