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 2 m + 1 , q ) \mathfrak {B}_n(-q^{2m+1},q) and the quantum algebra associated to the symplectic Lie algebra s p 2 m \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 Z [ q , q − 1 ] \mathbb {Z}[q,q^{-1}] -algebra, the quantized coordinate algebra defined by Kashiwara (which he denoted by A q Z ( g ) A_q^{\mathbb {Z}}(g) ) is isomorphic to the quantized coordinate algebra arising from a generalized Faddeev–Reshetikhin–Takhtajan construction.