$q$-deformed Howe duality for orthosymplectic Lie superalgebras

TL;DR

This paper develops a $q$-deformed version of Howe duality for orthosymplectic Lie superalgebras, establishing commuting actions on a supersymmetric space and recovering classical dualities in the limit.

Contribution

It introduces explicit commuting actions of quantized algebras and describes the semisimple decomposition, extending classical dualities to the $q$-deformed setting.

Findings

Established $q$-analogues of $(rak{g},G)$-dualities

Defined commuting actions of quantum groups on superspaces

Recovered classical dualities in the limit as $q$ approaches 1

Abstract

We give a $q$-analogue of Howe duality associated to a pair $(\mf{g},G)$, where $\mf{g}$ is an orthosymplectic Lie superalgebra and $G=O_\ell, Sp_{2\ell}$. We define explicitly {commuting actions} of a quantized enveloping algebra of $\mf{g}$ and the $\imath$quantum group of {type AI and AII} on a $q$-deformed supersymmetric space, and describe its semisimple decomposition whose classical limit recovers the $(\mf{g},G)$-duality. As special cases, we obtain $q$-analogues of $(\mf{g},G)$-dualities on symmetric and exterior algebras for $\mf{g}=\mf{so}_{2n}$, $\mf{sp}_{2n}$.