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

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 $\mathrm{Sym}(V^{\oplus r}\oplus V^{\ast \oplus s})$ 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 $\mathrm{Sym}(V^{\oplus r}\oplus V^{\ast \oplus s})$ 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 $\mathrm{Sym}(V^{\ast \oplus s})$. 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.

34 pages, 2 figures