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

TL;DR

This paper establishes an integral version of Schur--Weyl duality linking BMW algebras and quantum symplectic Lie algebras, extending the duality to arbitrary ground rings and connecting different quantized coordinate algebra constructions.

Contribution

It proves an integral Schur--Weyl duality for BMW algebras and quantum symplectic Lie algebras, answering a longstanding question and relating two quantized coordinate algebra frameworks.

Findings

Schur--Weyl duality holds over arbitrary ground rings.

BMW algebra and quantum symplectic algebra are dual in this setting.

Quantized coordinate algebras from different constructions are isomorphic.

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.