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 ${\mathrm{U}}_q({\mathfrak{gl}}_{m|n})$. A noncommutative ${\mathrm{U}}_q({\mathfrak{gl}}_{m|n})$-module superalgebra ${\mathcal{P}}_{r|s}^{k|l}$ 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 ${\mathrm{U}}_q({\mathfrak{gl}}_{m|n})$-invariants in ${\mathcal{P}}_{r|s}^{k|l}$ 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 ${\mathrm{U}}_q({\mathfrak{gl}}_{m|n})$. 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 ${\mathrm{U}}_q({\mathfrak{gl}}_{m|n})$. Applications to the quantum general linear group ${\mathrm{U}}_q({\mathfrak{gl}}_m)$ and the general linear superalgebra ${\mathfrak{gl}}_{m|n}$ will also be discussed.