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The classical invariant theory for the queer Lie superalgebra is aninvestigation of the $\mathrm{U}(\mathfrak{q}_n)$-invariant sub-superalgebra ofthe symmetric superalgebra $\mathrm{Sym}(V^{\oplus r}\oplus V^{*\oplus s})$ for$V=\mathbb{C}^{n|n}$. We establish the first fundamental theorem of invarianttheory for the quantum queer superalgebra $\mathrm{U}_q(\mathfrak{q}_n)$. Thekey ingredient is a quantum analog $\mathcal{O}_{r,s}$ of the symmetricsuperalgebra $\mathrm{Sym}(V^{\oplus r}\oplus V^{*\oplus s})$ that is createdas 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 tensorproduct is created via an explicit intertwining operator instead of theuniversal $\mathcal{R}$-matrix.

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