Quantum Howe duality and invariant polynomials

Abstract We construct two examples of q -deformed classical Howe dual pairs ( sl ( 2 , C ) , so ( 3 , C ) ) and ( sl ( 2 , C ) , sl ( n , C ) ) . Moreover, we obtain a noncommutative version of the first fundamental theorem of classical invariant theory. Our approach to these dualities differs from [9] and [11] . Furthermore, we solve the tensor product decomposition problem for Verma modules over U q ( sl ( 2 , C ) ) provided q is not a root of unity.