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Non-abelian extensions of infinite-dimensional Lie groups

Karl-Hermann Neeb (2007)

Annales de l’institut Fourier

In this article we study non-abelian extensions of a Lie group G modeled on a locally convex space by a Lie group N . The equivalence classes of such extension are grouped into those corresponding to a class of so-called smooth outer actions S of G on N . If S is given, we show that the corresponding set Ext ( G , N ) S of extension classes is a principal homogeneous space of the locally smooth cohomology group H s s 2 ( G , Z ( N ) ) S . To each S a locally smooth obstruction class χ ( S ) in a suitably defined cohomology group H s s 3 ( G , Z ( N ) ) S is defined....

Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds

Emmanuel Russ, Yannick Sire (2011)

Studia Mathematica

Let G be a real connected Lie group with polynomial volume growth endowed with its Haar measuredx. Given a C² positive bounded integrable function M on G, we give a sufficient condition for an L² Poincaré inequality with respect to the measure M(x)dx to hold on G. We then establish a nonlocal Poincaré inequality on G with respect to M(x)dx. We also give analogous Poincaré inequalities on Riemannian manifolds and deal with the case of Hardy inequalities.

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