The first and second fundamental theorems of invariant theory for the quantum general linear supergroup

Abstract

We develop the non-commutative polynomial version of the invariant theory for the quantum general linear supergroup ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$. A non-commutative ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-module superalgebra $\mathcal{P}^{k|l}_{\,r|s}$ is constructed, which is the quantum analogue of the supersymmetric algebra over $\mathbb{C}^{k|l}\otimes \mathbb{C}^{m|n}\oplus \mathbb{C}^{r|s}\otimes (\mathbb{C}^{m|n})^{\ast}$. We analyse the structure of the subalgebra of ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-invariants in $\mathcal{P}^{k|l}_{\,r|s}$ by using the quantum super analogue of Howe duality. The subalgebra of ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-invariants in $\mathcal{P}^{k|l}_{\,r|s}$ is shown to be finitely generated. We determine its generators and establish a surjective superalgebra homomorphism from a braided supersymmetric algebra onto it. This establishes the first fundamental theorem of invariant theory for ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$. We show that the above mentioned superalgebra homomorphism is an isomorphism if and only if $m\geq \min\{k,r\}$ and $n\geq \min\{l,s\}$, and obtain a monomial basis for the subalgebra of invariants in this case. When the homomorphism is not injective, we give a representation theoretical description of the generating elements of the kernel associated to the partition $((m+1)^{n+1})$, producing the second fundamental theorem of invariant theory for ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$. We consider two applications of our results. A complete treatment of the non-commutative polynomial version of invariant theory for ${\rm{ U}}_q(\mathfrak{gl}_{m})$ is obtained as the special case with $n=0$, where an explicit SFT is proved, which we believe to be new. The FFT and SFT of the invariant theory for the general linear superalgebra are recovered from the classical (i.e., $q\to 1$) limit of our results.