The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

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 $U(\mathfrak{g})$.