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

Jun Hu

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 ${\mathfrak{sp}}_{2m}$. In particular, we deduce that this Schur-Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang (J. Alg. 306 138–174) 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 (Duke Math. J. 69 455–485) (which was denoted by $A_q^{\mathbb{Z}}(g)$ there) is isomorphic to the quantized coordinate algebra arising from a generalized Faddeev-Reshetikhin-Takhtajan construction.

Keywords: Birman-Murakami-Wenzl algebra, modified quantized enveloping algebra, canonical bases.

AMS Subject Classification: Primary 17B37, 20C20; Secondary 20C08.