A convexity theorem for Poisson actions of compact Lie groups

The Z*-theorem for compact Lie groups

Glauberman’s classical Z∗-theorem is a theorem about involutions of finite groups (i.e. elements of order 2). It is one of the important ingredients for the classification of finite simple groups, which in turn allows to prove the corresponding theorem for elements of arbitrary prime order p. Let us recall the statement: if G is a finite group with a Sylow p-subgroup P , and if x is an element ...

Ergodic Actions of Semisimple Lie Groups on Compact Principal Bundles

Let G = SL(n, R) (or, more generally, let G be a connected, noncompact, simple Lie group). For any compact Lie group K, it is easy to find a compact manifold M , such that there is a volume-preserving, connection-preserving, ergodic action of G on some smooth, principal K-bundle P over M. Can M can be chosen independent of K? We show that if M = H / A is a homogeneous space, and the action of G...

Differentiable Actions of Compact Abelian Lie Groups on S»

Lie-poisson Structure on Some Poisson Lie Groups

Poisson Lie groups appeared in the work of Drinfel'd (see, e.g., [Drl, Dr2]) as classical objects corresponding to quantum groups. Going in the other direction, we may say that a Poisson Lie group is a group of symmetries of a phase space that are allowed to "twist," in a certain sense, the symplectic or Poisson structure. The Poisson structure on the group controls this twisting in a precise w...

Compact Lie Groups

The first half of the paper presents the basic definitions and results necessary for investigating Lie groups. The primary examples come from the matrix groups. The second half deals with representation theory of groups, particularly compact groups. The end result is the Peter-Weyl theorem.