# Yang Zhang (University of Sydney)

## Noncommutative classical invariant theory for the quantum general linear supergroup

We will give the noncommutative polynomial version of the invariant theory for the quantum general linear supergroup $${\rm{ U}}_q(\mathfrak{gl}_{m|n})$$. A noncommutative $${\rm{ U}}_q(\mathfrak{gl}_{m|n})$$-module superalgebra $$\mathcal{P}^{k|l}_{\,r|s}$$ is constructed, which is the quantum analogue of the supersymmetric algebra over $$\mathbb{C}^{k|l}\otimes \mathbb{C}^{m|n}\oplus \mathbb{C}^{r|s}\otimes (\mathbb{C}^{m|n})^{\ast}$$. 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 (FFT) of invariant theory for $${\rm{ U}}_q(\mathfrak{gl}_{m|n})$$. When the above homomorphism is not injective, we give a representation theoretical description of the generating elements of the kernel associated to the partition $$((m+1)^{n+1})$$, which amounts to the second fundamental theorem (SFT) of invariant theory for $${\rm{ U}}_q(\mathfrak{gl}_{m|n})$$. Applications to the quantum general linear group $${\rm{ U}}_q(\mathfrak{gl}_{m})$$ and the general linear superalgebra $$\mathfrak{gl}_{m|n}$$ will also be discussed.