Twisted Lie group C-algebras as strict quantizations

A nonzero 2-cocycle Γ ∈ Z(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 ∈ Z (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 ~ assumes values in (Z\{0}) . This means that there exists a continuous field of C∗-algebras, indexed by ~ ∈ 0 ∪ (Z\{0}), for which A = C0(g ∗) and A = C∗(G, c) for ~ 6= 0, along with a cross-section of the field satisfying Dirac’s condition asymptotically relating the commutator in A to the Poisson bracket on C∞(g∗(Γ)). Note that the ‘quantization’ of ~ does not occur for Γ = 0. ∗Supported by a fellowship from the Royal Netherlands Academy of Arts and Sciences (KNAW)