Twisted Lie Group C*-Algebras as Strict Quantization
Twisted Lie Group C*-Algebras as Strict Quantization
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作为严格量化的扭曲李群 C*-代数
DOI：
10.1023/a:1007525214561
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发表时间：
1998
影响因子：
1.2
通讯作者：
N. P. Landsman
中科院分区：
文献类型：
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作者：
N. P. Landsman
紧李群G的李代数g上的一个非零2-上循环Γ∈ Z2（g，R）定义了对偶李代数g* 上的一个扭曲版本的Lie-Poisson结构，从而得到Poisson代数C∞（g*（Γ））.类似地，在单位元附近光滑的G上的乘数c∈ Z2（G，U（1））定义了G上卷积积中的扭曲，由扭曲群C-代数C*（G，c）编码。在C∞（g*（r））和C*（G，c）之间进行了一些肤浅但有启发性的类比之后，证明了后者是前者的严格量子化，其中Planck常数预言假定值为（Z 0）-1。这意味着存在一个C*-代数的连续域，索引为∈ 0 ‡（Z\{0}）-1，其中A0= C 0（g*）且A =C*（G，c），≠ 0，沿着满足狄拉克条件的域的横截面渐进地将A中的换位子与C∞（g*（r））上的Poisson括号联系起来。注意，预测的“量化”对于r =0不发生。
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.