## 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.

: Primary 17B37, 20C20; Secondary 20C08.

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 Friday, December 11, 2009