1998/07/24 by N.P. Landsman, N. P. Landsman, Landsman, N. P. · 2 citations
Mathematics · Physics and Astronomy · #81S10 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #Symplectic Geometry (math.SG) #math-ph #math.MP #math.OA #math.SG #msc:81S10
paper · pdf · doi:10.48550/arxiv.math-ph/9807028
7 pages
arxiv created 1998/07/24 · openalex publication_date 1998/07/24 · arxiv updated 2016/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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\backslash\0\)-1. This means that there exists a continuous field of C^*-algebras, indexed by ℏ∈ 0∪ (\Z\backslash\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.