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Register a new userNonabelian cohomology of compact Lie groups
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Given a Lie group $G$ with finitely many components and a compact Lie group A which acts on $G$ by automorphisms, we prove that there always exists an A-invariant maximal compact subgroup K of G, and that for every such K, the natural map $H^1(A,K)to H^1(A,G)$ is bijective. This generalizes a classical result of Serre [6] and a recent result in [1].
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In this paper we prove some properties of the nonabelian cohomology $H^1(A,G)$ of a group $A$ with coefficients in a connected Lie group $G$. When $A$ is finite, we show that for every $A$-submodule $K$ of $G$ which is a maximal compact subgroup of $G$, the canonical map $H^1(A,K)to H^1(A,G)$ is bijective. In this case we also show that $H^1(A,G)$ is always finite. When $A=ZZ$ and $G$ is compact, we show that for every maximal torus $T$ of the identity component $G_0^ZZ$ of the group of invariants $G^ZZ$, $H^1(ZZ,T)to H^1(ZZ,G)$ is surjective if and only if the $ZZ$-action on $G$ is 1-semisimple, which is also equivalent to that all fibers of $H^1(ZZ,T)to H^1(ZZ,G)$ are finite. When $A=Zn$, we show that $H^1(Zn,T)to H^1(Zn,G)$ is always surjective, where $T$ is a maximal compact torus of the identity component $G_0^{Zn}$ of $G^{Zn}$. When $A$ is cyclic, we also interpret some properties of $H^1(A,G)$ in terms of twisted conjugate actions of $G$.
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
We prove continuity results for abstract epimorphisms of locally compact groups onto finitely generated groups.
Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $mathrm{Hom}$-$otimes$ identities associated to the rational discrete bimodule $mathrm{Bi}(G)$ allow to introduce the notion of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretins group of spheromorphisms of a locally finite regular tree is not even of finite rational discrete cohomological dimension. For a unimodular t.d.l.c. group $G$ of type $mathrm{FP}$ it is possible to define an Euler-Poincare characteristic $chi(G)$ which is a rational multiple of a Haar measure. This value is calculated explicitly for Chevalley groups defined over a non-discrete, non-archimedean local field $K$ and some other examples.
A generalized Euler parameterization of a compact Lie group is a way for parameterizing the group starting from a maximal Lie subgroup, which allows a simple characterization of the range of parameters. In the present paper we consider the class of all compact connected Lie groups. We present a general method for realizing their generalized Euler parameterization starting from any symmetrically embedded Lie group. Our construction is based on a detailed analysis of the geometry of these groups. As a byproduct this gives rise to an interesting connection with certain Dyson integrals. In particular, we obtain a geometry based proof of a Macdonald conjecture regarding the Dyson integrals correspondent to the root systems associated to all irreducible symmetric spaces. As an application of our general method we explicitly parameterize all groups of the class of simple, simply connected compact Lie groups. We provide a table giving all necessary ingredients for all such Euler parameterizations.
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