$q$-deformed Howe duality for orthosymplectic Lie superalgebras

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 $G=O_{\ell },S{p}_{2\ell }$. We define explicitly {commuting actions} of a quantized enveloping algebra of $\mf{g}$ and the $ı$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 $\mf{g}=\mf{so}_{2n}$, $\mf{sp}_{2n}$.

37 pages, more details are provided for the proofs of Propositions 3.4 and 4.1, the presentation of Section 4.1 has been revised in a more simplified way, the proof of Lemma 4,6 is revised and clarified, to appear in Letters in Mathematical Physics