Quantum Howe duality and invariant polynomials

TL;DR

This paper develops q-deformed Howe dual pairs, establishes a noncommutative invariant theory, and solves tensor product decompositions for Verma modules over quantum sl(2,C).

Contribution

It introduces new q-deformed dual pairs and extends classical invariant theory into a noncommutative setting, differing from previous approaches.

Findings

Constructed q-deformed Howe dual pairs (sl(2,C), sl(2,C)) and (sl(2,C), sl(n,C))

Established a noncommutative version of the first fundamental theorem of invariant theory

Solved tensor product decomposition for Verma modules over U_q(sl(2,C)) when q is not a root of unity

Abstract

We construct two examples of q-deformed classical Howe dual pairs (sl(2,C), sl(2,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 duality differs from the paper of Lehrer-Zhang-Zhang. 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.