First fundamental theorems of invariant theory for quantum supergroups

Abstract

Let \(\mathrm{U}_q({{\mathfrak {g}}})\) be the quantum supergroup of \({\mathfrak {gl}}_{m|n}\) or the modified quantum supergroup of \({\mathfrak {osp}}_{m|2n}\) over the field of rational functions in q, and let \(V_q\) be the natural module for \(\mathrm{U}_q({{\mathfrak {g}}})\). There exists a unique tensor functor associated with \(V_q\), from the category of ribbon graphs to the category of finite dimensional representations of \(\mathrm{U}_q({{\mathfrak {g}}})\), which preserves ribbon category structures. We show that this functor is full in the cases \({{\mathfrak {g}}}={\mathfrak {gl}}_{m|n}\) or \({\mathfrak {osp}}_{2\ell +1|2n}\). For \({{\mathfrak {g}}}={\mathfrak {osp}}_{2\ell |2n}\), we show that the space 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 \(\mathrm{U}({{\mathfrak {g}}})\).