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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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##### References:
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