Twisted Lie Group C*-Algebras as Strict Quantization

Abstract

A nonzero 2-cocycle Γ∈ Z2(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∞ (g*(Γ)). Similarly, a multiplier c∈ Z2(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∞ (g*(Γ)) and C*(G,c), it is shown that the latter is a strict quantization of the former, where Planck’s constant ħ assumes values in (Z\{0})-1. This means that there exists a continuous field of C*-algebras, indexed by ħ ∈ 0 ∪ (Z\{0})-1, for which A0= C0(g*) and Aħ=C*(G,c) for ħ ≠ 0, along with a cross-section of the field satisfying Dirac’s condition asymptotically relating the commutator in Aħ to the Poisson bracket on C∞(g*(Γ)). Note that the ‘quantization’ of ħ does not occur for Γ=0.