Braided tensor products and polynomial invariants for the quantum queer superalgebra

Abstract

The classical invariant theory for the queer Lie superalgebra ${\mathfrak{q}}_n$ investigates its invariants in the supersymmetric algebra $${\mathcal{U}}_{s,l}^{r,k}:=\mathrm{Sym}\left(V^{\oplus r}\oplus \Pi (V{)}^{\oplus k}\oplus V^{\ast \oplus s}\oplus \Pi (V^{\ast }{)}^{\oplus l}\right),$$ where $V={\mathbb{C}}^{n|n}$ is the natural supermodule, $V^{\ast }$ is its dual and $\Pi$ is the parity reversing functor. This paper aims to construct a quantum analogue ${\mathcal{B}}_{s,l}^{r,k}$ of ${\mathcal{U}}_{s,l}^{r,k}$ and to explore the quantum queer superalgebra ${\mathrm{U}}_q({\mathfrak{q}}_n)$-invariants in ${\mathcal{B}}_{s,l}^{r,k}$. The strategy involves braided tensor products of the quantum analogues ${\mathsf{A}}_{r,n}$, ${\mathsf{A}}_{k,n}^{\Pi }$ of the supersymmetric algebras $\mathrm{Sym}\left(V^{\oplus r}\right)$, $\mathrm{Sym}\left(\Pi (V{)}^{\oplus k}\right)$, and their dual partners ${\bar{\mathsf{A}}}_{s,n}$, and ${\bar{\mathsf{A}}}_{l,n}^{\Pi }$. These braided tensor products are defined using explicit braiding operator due to the absence of a universal R-matrix for ${\mathrm{U}}_q({\mathfrak{q}}_n)$. Furthermore, we obtain an isomorphism between the braided tensor product ${\mathsf{A}}_{r,n}\otimes {\mathsf{A}}_{k,n}$ and ${\mathsf{A}}_{r+k,n}$, an isomorphism between ${\mathsf{A}}_{k,n}^{\Pi }$ and ${\mathsf{A}}_{k,n}$, as well as the corresponding isomorphisms for their dual parts. Consequently, the ${\mathrm{U}}_q({\mathfrak{q}}_n)$-supermodule superalgebra ${\mathcal{B}}_{s,l}^{r,k}$ is identified with ${\mathcal{B}}_{s+l,0}^{r+k,0}$. This allows us to obtain a set of generators of ${\mathrm{U}}_q({\mathfrak{q}}_n)$-invariants in ${\mathcal{B}}_{s,l}^{r,k}$.