## University of Sydney Algebra Seminar

# Yang Zhang (University of Sydney)

## Friday 9 June, 12-1pm, Place: Carslaw 375

## 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.