The classical invariant theory for the queer Lie superalgebra \({\mathfrak{q}}_n\) > investigates its invariants in the supersymmetric algebra \({\mathscr{U}}_{s,l}^{r,k}:=\text{Sym}(V^{\bigoplus r}\bigoplus \Pi(V)^{\bigoplus k}\bigoplus V^{*\bigoplus s}\bigoplus \Pi(V^{*})^{\bigoplus l}),\) where V = ℂn∣n is the natural module of \(\mathfrak{q}_{n}\) , V* is its dual and Π is the parity reversing functor. This paper aims to construct a quantum analogue \({\mathscr{B}}_{s,l}^{r,k}\) of \({\mathscr{U}}_{s,l}^{r,k}\) and to explore the quantum queer superalgebra \(\mathrm{U}_{q}(\mathfrak{q}_{n})\) -invariants in \({\mathscr{B}}_{s,l}^{r,k}\) .
The strategy involves braided tensor products of the quantum analogues Ar,n, A k,n Π of the supersymmetric algebras Sym(V⊕r), Sym(Π(V)⊕k), and their dual partners Ās,n, and Ā l,n Π . These braided tensor products are defined using explicit braiding operators 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 Ar,n ⊗ Ak,n and Ar+k,n, an isomorphism between A k,n Π and Ak,n, as well as the corresponding isomorphisms for their dual parts. Consequently, the \(\mathrm{U}_{q}(\mathfrak{q}_{n})\) -module superalgebra \({\mathscr{B}}_{s,l}^{r,k}\) is identified with \({\mathscr{B}}_{s+l,0}^{r+k,0}\) . This allows us to obtain a set of generators of \(\mathrm{U}_{q}(\mathfrak{q}_{n})\) -invariants in \({\mathscr{B}}_{s,l}^{r,k}\) .
