Howe Duality for Quantum Queer Superalgebras

Abstract

We establish a new Howe duality between a pair of quantum queer superalgebras $({\mathrm{U}}_{q^{-1}}({\mathfrak{q}}_n),{\mathrm{U}}_q({\mathfrak{q}}_m))$. The key ingredient is the construction of a non-commutative analogue ${\mathcal{A}}_q({\mathfrak{q}}_n,{\mathfrak{q}}_m)$ of the symmetric superalgebra $S({\mathbb{C}}^{mn|mn})$ with the use of quantum coordinate queer superalgebra. It turns out that this superalgebra is equipped with a ${\mathrm{U}}_{q^{-1}}({\mathfrak{q}}_n)\otimes {\mathrm{U}}_q({\mathfrak{q}}_m)$-supermodule structure that admits a multiplicity-free decomposition. We also show that the $({\mathrm{U}}_{q^{-1}}({\mathfrak{q}}_n),{\mathrm{U}}_q({\mathfrak{q}}_m))$-Howe duality implies the Sergeev-Olshanski duality.