On the Quantum Metaplectic Howe Duality

TL;DR

This paper develops a quantum version of the classical Howe duality involving symplectic and special linear Lie algebras, introducing new $q$-analogues of key operators and decompositions in the context of quantum groups.

Contribution

It establishes a quantum analogue of the classical metaplectic Howe duality for the case n=1, including $q$-deformed operators and decompositions.

Findings

Constructed commuting representations of quantum groups $( ext{U}_q( ext{sl}_2), ext{U}_{q^2}( ext{sl}_2))$

Derived $q$-analogues of the symplectic Dirac operator and Fischer decomposition

Provided explicit formulas for projections onto symplectic polynomial monogenics

Abstract

We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras $(\mathfrak{sp}_{2n},\mathfrak{sl}_2)$ in the case when $n=1$. Our results yield commuting representations of the pair of Drinfeld-Jimbo quantum groups $(\mathcal U_{q^2}(\mathfrak{sl}_2),\mathcal U_{q}(\mathfrak{sl}_2))$ realized in a suitable algebra of $q$-differential operators acting on the space of symplectic polynomial spinors. We obtain $q$-analogues for the symplectic Dirac operator, the Fischer decomposition, the expression for the symplectic polynomial monogenics and for the projection operators onto the monogenics. We also discuss $q$-analogues of generalized symmetries of the $q$-symplectic Dirac operator raising the homogeneous polynomial degree.