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Quantum groupoids of the finite type and quantization on orbit spaces

Heller M., Odrzygóźdź Z., Pysiak L., Sasin W.

Demonstratio Mathematica

2004

Vol. 37, nr 3

671--678

We show that the Hopf algebra on a transformation groupoid F = E x G where G is a finite group acting on the total space of a principal fibre boundle over M = E/G, is the cross product of the algebras C°°(E) and CG. We study duality properties of this algebra, and consider quantization on orbit spaces program in this context.

Union of equivariant extensors and equivariant covering spaces

Jimenez R., Neymet S.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 4

347-356

Let G be a compact topological group and P be the class of all paracompact and Hausdorff G-spaces. We prove that a paracompact space that is a local G-AN E(P) is a G-AN E(P). Then we give an application to the theory of equivariant overlays by proving the following: Let p : [...] --> X be G-overlays of a metrizable connected G-space X. Then an equivariant map f : Z --> X from Z, a metrizable connected G-space, can be lifted to an equivariant map F : Z --> [...] if and only if f[...].

TheBanach-Mazur compacta are absolute retracts

Antonian S.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 2

113-119

We prove that the space N(E) of all norms on n-dimensional linear space E is a proper G-space under the natural action of the full linear group G = GL(n). The Banach-Mazur compactum Q(n) is just the orbit space of the G-space N(E). We show here that for any n [is greater than or equal to] 1, Q(n) is an absolute retract for metric spaces. We generalize this fact be showing that for a locally compact almost connected group G, its maximal compact subgroup K, and a proper G-space X, X/G is an A(N)R space provided it is metrizable and X is a K-A(N)R space.