Differential geometry and quantization on a locally compact group

摘要

For an arbitrary locally compact group G, we describe the structure of the Lie algebra X(G) of vector fields, the exterior algebra A(G) of differential forms, and the Poisson algebra of symbols on G polynomial with respect to the momenta. A continuous left-invariant gp-quantizaton is constructed, giving rise to a one-to-one correspondence between symbols and differential operators on G. It is demonstrated that neither of the other two classical quantizations, namely, the pq and Weyl quantizations, can be constucted on an infinite group G if the same properties are to be retained.