First fundamental theorems of invariant theory for quantum supergroups

Abstract

Let $U_q(\mathfrak{g})$ be the quantum supergroup of ${\mathfrak{gl}}_{m|n}$ or the modified quantum supergroup of $os{p}_{m|2n}$ 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 $os{p}_{2\ell +1|2n}$. For $\mathfrak{g}=os{p}_{2\ell |2n}$, we show that the space $Ho{m}_{U_q(\mathfrak{g}}(V_q^{\otimes r},V_q^{\otimes s})$ 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 $U(\mathfrak{g})$.