Quantization of Poisson algebras associated to Lie algebroids (Q2785090)

After an informative introduction to quantization theory in general, the authors begin their study of the subject of the title with a careful series of definitions of the principle structures that they need: Lie groupoids and algebroids, the exponential map(s) connecting the two, the \(C^*\)-algebra and reduced \(C^*\)-algebra of a Lie groupoid, and the associated Haar (measure) systems. For quantization theory more particularly, they then introduce what they mean by a Poisson algebra and manifold, and the Kirchberg-Wassermann definition of Dixmier's notion of a field of \(C^*\)-algebras. NEWLINENEWLINENEWLINEThe basic type of quantization the authors consider is deformation quantization, in which the `quantum' aspect is expressed through a one-parameter field of \(C^*\)-algebras, and the `classical' aspect by a Poisson algebra identified with the parameter value \(0\). This notion of an abrupt change of structural type at a special value of a parameter is not restricted to quantization: it appears in the theory of group contractions, for example, and deformation theory would seem to be a natural formalism to describe it. In the case of quantization, this specialization is connected to the mode in which the limit of the quantum commutation relations converge to commuting quantities, hence the term ``strict quantization''. NEWLINENEWLINENEWLINEThe principal results of this paper are the existence of a strict quantization associated with the generic Poisson manifold of a Lie groupoid, and the corresponding semi-strict quantization associated with the Poisson manifold of a Lie algebroid of a Lie groupoid. NEWLINENEWLINENEWLINEEven stating these results in not very technical terms indicates that this is necessarily a highly technical result involving the confluence of operator theory and differential geometry. The second half of the paper is devoted to the proof, including a generalization of the Fourier transform on Schwartz space to a vector bundle analogue. NEWLINENEWLINENEWLINEThe paper ends with some examples of quantization: over a dual Lie algebra, a transformation group \(C^*\)-algebra and the standard Weyl quantization for \(n\) (flat) degrees of freedom. Finally, the authors note a connection between Atiyah-Singer index theory, \(K\)-theory and quantization, and remark that the general results of this paper might yield a generalization of these results as well.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00029].