Twisted Lie group $C^*$-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 $C^*$-algebra C^*(G,c)$. Further to some superficial yet enlightening analogies between $C^{\infty}(\g^*_{(Γ)})$ and $C^*(G,c)$, it is shown that the latter is a strict quantization of the former, where Planck's constant $\hbar$ assumes values in $(\Z\backslash\{0\})^{-1}$. This means that there exists a continuous field of $C^*$-algebras, indexed by $\hbar\in 0\cup (\Z\backslash\{0\})^{-1}$, for which $\A^0=C_0(\g^*)$ and $\A_{\hbar}=C^*(G,c)$ for $\hbar\neq 0$, along with a cross-section of the field satisfying Dirac's condition asymptotically relating the commutator in $\A_{\hbar}$ to the Poisson bracket on $C^{\infty}(\g^*_{(Γ)})$. Note that the `quantization' of $\hbar$ does not occur for $Γ=0$.