Howe Duality for Quantum Queer Superalgebras

TL;DR

This paper establishes a new Howe duality between quantum queer superalgebras, introducing a non-commutative superalgebra with a multiplicity-free module structure, and connects it to Sergeev-Olshanski duality.

Contribution

It constructs a non-commutative analogue of the symmetric superalgebra for quantum queer superalgebras and proves a new Howe duality that implies Sergeev-Olshanski duality.

Findings

Constructed a non-commutative superalgebra with supermodule structure.

Proved the multiplicity-free decomposition under the duality.

Connected Howe duality to Sergeev-Olshanski duality.

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.