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Symplectic models of groups with noncommutative spaces. (English) Zbl 0841.58006
Bureš, J. (ed.) et al., The proceedings of the Winter school Geometry and topology, Srní, Czechoslovakia, January 1992. Palermo: Circolo Matemático di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 32, 185-194 (1994).
In the setting of \(C^*\)-algebras, noncommutative geometry is understood as viewing noncommutative \(C^*\)-algebras as generalizations of locally compact spaces. The objects in this category which generalize locally compact groups are then quantum groups in the sense of Woronowicz. From the point of view of physics these generalizations are well adapted to describe symmetries of quantum systems, but they cannot describe symmetries of classical systems.
In this paper, the author gives an overview on an alternative generalization of spaces and groups, which is based on symplectic geometry and thus rather adapted to classical systems. Basically, he defines a symplectic \(*\)-algebra or \(S^*\)-space as a symplectic manifold with three symplectic relations, which generalize the fiberwise additive structure on a cotangent bundle. Next, he outlines the definition of \(S^*\)-groups, referring to his papers Commun. Math. Phys. 134, No. 2, 347-370 (1990; Zbl 0708.58030) and ibid., 371-395 (1990; Zbl 0708.58031) for the precise definition. Finally, he discusses a relation between \(S^*\)-spaces and Poisson manifolds and relations between \(S^*\)-groups, Manin groups, Poisson Lie groups and Lie bialgebras.
For the entire collection see [Zbl 0794.00022].
Reviewer: A.Cap (Wien)
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B37 Quantum groups (quantized enveloping algebras) and related deformations
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)