Braided tensor products and polynomial invariants for the quantum queer superalgebra

TL;DR

This paper constructs quantum analogues of classical invariant algebras for the queer Lie superalgebra using braided tensor products and explicit braiding, identifying invariants in the quantum setting.

Contribution

It introduces a quantum analogue of classical invariant algebras for $rak{q}_n$ and develops methods to analyze invariants using braided tensor products without a universal R-matrix.

Findings

Constructed quantum analogues of classical invariant algebras.

Established isomorphisms simplifying the structure of quantum invariants.

Derived generators for the quantum invariants in the superalgebra.

Abstract

The classical invariant theory for the queer Lie superalgebra $\mathfrak{q}_n$ investigates its invariants in the supersymmetric algebra $$\mathcal{U}_{s,l}^{r,k}:=\mathrm{Sym}\left(V^{\oplus r}\oplus \Pi(V)^{\oplus k}\oplus V^{*\oplus s}\oplus \Pi(V^*)^{\oplus l} \right),$$ where $V=\mathbb{C}^{n|n}$ is the natural supermodule, $V^*$ is its dual and $\Pi$ is the parity reversing functor. This paper aims to construct a quantum analogue $\mathcal{B}^{r,k}_{s,l}$ of $\mathcal{U}_{s,l}^{r,k}$ and to explore the quantum queer superalgebra $\mathrm{U}_q(\mathfrak{q}_n)$-invariants in $\mathcal{B}^{r,k}_{s,l}$. 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 $\mathcal{B}^{r,k}_{s,l}$ is identified with $\mathcal{B}^{r+k,0}_{s+l,0}$. This allows us to obtain a set of generators of $\mathrm{U}_q(\mathfrak{q}_n)$-invariants in $\mathcal{B}^{r,k}_{s,l}$.