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BMW algebra, quantized coordinate algebra and type \(C\) Schur–Weyl duality. (English) Zbl 1219.17008

Summary: We prove an integral version of the Schur-Weyl duality between the specialized Birman-Murakami-Wenzl algebra \( \mathfrak{B}_n(-q^{2m+1},q)\) and the quantum algebra associated to the symplectic Lie algebra \( \mathfrak{sp}_{2m}\). In particular, we deduce that this Schur-Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang in the symplectic case. As a byproduct, we show that, as a \( \mathbb{Z}[q,q^{-1}]\)-algebra, the quantized coordinate algebra defined by Kashiwara (which he denoted by \( A_q^{\mathbb{Z}}(g))\) is isomorphic to the quantized coordinate algebra arising from a generalized Faddeev-Reshetikhin-Takhtajan construction.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
20C20 Modular representations and characters
20C08 Hecke algebras and their representations
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