Algebra Seminar: Zhang -- Noncommutative classical invariant theory for the quantum general linear supergroup

Yang Zhang (University of Sydney) 

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

Title: Noncommutative classical invariant theory for the quantum general linear
supergroup 

Abstract: We will give the noncommutative polynomial version of the invariant theory for
the quantum general linear supergroup U_q(gl_{m|n}).  A noncommutative
U_q(gl_{m|n})-module superalgebra P^{k|l}_{r|s} is constructed, which is the quantum
analogue of the supersymmetric algebra over C^{k|l}\otimes C^{m|n}\oplus C^{r|s}\otimes
(C^{m|n})*.  The subalgebra of U_q(gl_{m|n})-invariants in 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 U_q(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 U_q(gl_{m|n}).
Applications to the quantum general linear group U_q(gl_m) and the general linear
superalgebra gl_{m|n} will also be discussed.