The first fundamental theorem of invariant theory for the quantum queer superalgebra

TL;DR

This paper proves the first fundamental theorem of invariant theory for the quantum queer superalgebra, establishing invariant sub-superalgebras using a novel braided tensor product construction without a universal R-matrix.

Contribution

It introduces a quantum analog of the symmetric superalgebra and establishes invariant theory results for the quantum queer superalgebra, overcoming the lack of a quasi-triangular structure.

Findings

Established the first fundamental theorem for quantum queer superalgebra invariants.

Constructed a quantum symmetric superalgebra using a braided tensor product.

Provided explicit intertwining operators for the construction.

Abstract

The classical invariant theory for the queer Lie superalgebra is an investigation of the $\mathrm{U}(\mathfrak{q}_n)$-invariant sub-superalgebra of the symmetric superalgebra $\mathrm{Sym}(V^{\oplus r}\oplus V^{*\oplus s})$ for $V=\mathbb{C}^{n|n}$. We establish the first fundamental theorem of invariant theory for the quantum queer superalgebra $\mathrm{U}_q(\mathfrak{q}_n)$. The key ingredient is a quantum analog $\mathcal{O}_{r,s}$ of the symmetric superalgebra $\mathrm{Sym}(V^{\oplus r}\oplus V^{*\oplus s})$ that is created as a braided tensor product of a quantization $\mathsf{A}_{r,n}$ of $\mathrm{Sym}(V^{\oplus r})$ and a quantization $\bar{\mathsf{A}}_{s,n}$ of $\mathrm{Sym}(V^{*\oplus s})$. Since the quantum queer superalgebra $\mathrm{U}_q(\mathfrak{q}_n)$ is not quasi-triangular, our braided tensor product is created via an explicit intertwining operator instead of the universal $\mathcal{R}$-matrix.