Quantized Weyl algebras, the double centralizer property, and a new first fundamental theorem for Uq(gln)

Abstract

Let P:=Pm×n denote the quantized coordinate ring of the space of m × n matrices, equipped with natural actions of the quantized enveloping algebras Uq(glm) and Uq(gln) . Let L and R denote the images of Uq(glm) and Uq(gln) in End(P) , respectively. We define a q-analogue of the algebra of polynomial-coefficient differential operators inside End(P) , henceforth denoted by PD , and we prove that L∩PD and R∩PD are mutual centralizers inside PD . Using this, we establish a new First Fundamental theorem of invariant theory for Uq(gln) . We also compute explicit formulas in terms of q-determinants for generators of the algebras Lh∩PD and Rh∩PD , where Lh and Rh denote the images of the Cartan subalgebras of Uq(glm) and Uq(gln) in End(P) , respectively. Our algebra PD and the algebra Pol(Matm,n)q that is defined in (Shklyarov et al 2004 Int. J. Math. 15 855–94) are related by extension of scalars, but we give a new construction of PD using deformed twisted tensor products.