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Howe duality for quantum queer superalgebras. (English) Zbl 1461.17015
Summary: We establish a new Howe duality between a pair of quantum queer superalgebras \(( {U}_{q^{- 1}}(\mathfrak{q}_n), {U}_q(\mathfrak{q}_m))\). The key ingredient is the construction of a non-commutative analogue \(\mathcal{A}_q(\mathfrak{q}_n, \mathfrak{q}_m)\) of the symmetric superalgebra \(\mathbb{S}(\mathbb{C}^{m n | m n})\) with the use of quantum coordinate queer superalgebra. It turns out that this superalgebra is equipped with a \({U}_{q^{- 1}}(\mathfrak{q}_n) \otimes {U}_q(\mathfrak{q}_m)\)-supermodule structure that admits a multiplicity-free decomposition. We also show that the \(({U}_{q^{- 1}}(\mathfrak{q}_n), {U}_q(\mathfrak{q}_m))\)-Howe duality implies the Sergeev-Olshanski duality.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G42 Quantum groups (quantized function algebras) and their representations
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