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Lie Group Quantum Mathematics Seminar

Quantum Weyl algebras and a new First Fundamental Theorem of invariant theory for U_q(gl(n))

Hadi Salmasian, University of Ottawa

Location:  Zoom
Date & time: Friday, 10 February 2023 at 12:10PM - 1:10PM

Abstract The First Fundamental Theorem (FFT) is one of the highlights of invariant theory of reductive groups and goes back to the works of Schur, Weyl, Brauer, etc. For the group GL(V), the FFT describes generators for polynomial invariants on direct sums of several copies of the standard module V and its dual V*. Then R. Howe pointed out that the FFT is closely related to a double centralizer statement inside a Weyl algebra (a.k.a. the algebra of polynomial-coefficient differential operators).

In this talk we first construct a quantum Weyl algebra and then present a quantum analogue of the FFT. We explain the relation between the FFT and a double centralizer property inside the quantum Weyl algebra. We remark that the FFT that we obtain is different from the one proved by G. Lehrer, H. Zhang, and R.B. Zhang for U_q(gl(n)). Time permitting, I will explain the connection between this work and the Capelli Eigenvalue Problem in the context of quantum symmetric spaces. This talk is based on a joint project with Gail Letzter and Siddhartha Sahi.

Zoom link https://rutgers.zoom.us/j/93921465287
Meeting ID: 939 2146 5287
Passcode: 196884, the dimension of the weight 2 homogeneous
subspace of the moonshine module

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