BMW algebra, quantized coordinate algebra and type C Schur--Weyl duality

Abstract

We prove an integral version of the Schur--Weyl duality between the specialized Birman--Murakami--Wenzl algebra \(B_n(-q^{2m+1},q)\) and the quantum algebra associated to the symplectic Lie algebra 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 [Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra (1) 306 (2006), 138--174] in the symplectic case. As a byproduct, we show that, as \(Z[q,q^{-1}]\)-algebra, the quantized coordinate algebra defined by Kashiwara is isomorphic to the quantized coordinate algebra arising from a generalized Faddeev--Reshetikhin--Takhtajan's construction.