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

Let P := P (m x n) denote the quantized coordinate ring of the space of m x 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 boolean AND P D and R boolean AND 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 boolean AND P D and R h boolean AND 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.