Twisted Lie group $C^{\ast }$-algebras as strict quantizations

Abstract

A nonzero 2-cocycle $Γ\in Z^2(\g,\R)$ on the Lie algebra $\g$ of a compact Lie group $G$ defines a twisted version of the Lie-Poisson structure on the dual Lie algebra $\g^*$, leading to a Poisson algebra $C^{\infty}(\g_{(Γ)}^*)$. Similarly, a multiplier $c\in Z^2(G,U(1))$ on $G$ which is smooth near the identity defines a twist in the convolution product on $G$, encoded by the twisted group $C^{\ast }$-algebra C^*(G,c)$.Furthertosomesuperficialyetenlighteninganalogiesbetween$C^{\infty}(\g^*_{(Γ)})$and$C^*(G,c)$,itisshownthatthelatterisastrictquantizationoftheformer,wherePlanc{k}^{\prime }sconstant$\hbar$assumesvaluesin$(\Z\backslash\{0\})^{-1}$.Thismeansthatthereexistsacontinuousfieldof$C^*$-algebras,indexedby$\hbar\in 0\cup (\Z\backslash\{0\})^{-1}$,forwhich$\A^0=C_0(\g^*)$and$\A_{\hbar}=C^*(G,c)$for$\hbar\neq 0$,alongwithacross-sectionofthefieldsatisfyingDira{c}^{\prime }sconditionasymptoticallyrelatingthecommutatorin$\A_{\hbar}$tothePoissonbracketon$C^{\infty}(\g^*_{(Γ)})$.Notethatthe‘quantizatio{n}^{\prime }of$\hbar$doesnotoccurfor$Γ=0$.

7 pages

Cited by in corpus (5)

• Lie groupoid C*-algebras and Weyl quantization
• Deformation quantization and the Baum-Connes conjecture
• Extensions of Bundles of C*-algebras
• Classical Limits of Unbounded Quantities by Strict Quantization
• Classical Limits of Hilbert Bimodules as Symplectic Dual Pairs