## 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.