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

Let P := Pm×n denote the quantized coordinate ring of the space of m × n matrices. We introduce a q-analog PD of the algebra of polynomial coefficient differential operators inside End(P) and we prove that PD is an integral form of the algebra Pol(Matm×n)q introduced earlier by Shklyarov-Sinel'shchikov-Vaksman [SSV04]. Let L and R denote the images in End(P) of the natural actions of the quantized enveloping algebras Uq(glm) and Uq(gln), respectively, and let Lh and Rh denote the images of their Cartan subalgebras. Our main result is that L ∩ PD and R ∩ PD are mutual centralizers in PD, and using this, we establish a new First Fundamental Theorem of invariant theory for Uq(gln). We also determine explicit generators of the subalgebras Lh ∩ PD and Rh ∩ PD in terms of q-determinants.