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

Let P := P m × n denote the quantized coordinate ring of the space of m × n matrices, equipped with natural actions of the quantized enveloping algebras U q ( gl m ) and U q ( gl n ) . Let L and R denote the images of U q ( gl m ) and U q ( gl n ) in End ( P ) , respectively. We define a q -analogue of the algebra of polynomial-coefficient differential operators inside End ( P ) , henceforth denoted by P D , and we prove that L ∩ P D and R ∩ P D are mutual centralizers inside P D . Using this, we establish a new First Fundamental theorem of invariant theory for U q ( gl n ) . We also compute explicit formulas in terms of q -determinants for generators of the algebras L h ∩ P D and R h ∩ P D , where L h and R h denote the images of the Cartan subalgebras of U q ( gl m ) and U q ( gl n ) in End ( P ) , respectively. Our algebra P D and the algebra Pol ( Mat m , 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 P D using deformed twisted tensor products.