  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.
