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Register a new userNonabelian cohomology with coefficients in Lie groups
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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 $G$.
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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].
We obtain non-vanishing of group $L^p$-cohomology of Lie groups for $p$ large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromovs question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for $L^p$-cohomology, allowing the use of quasi-isometry invariance.
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.
The topological classification of gerbes, as principal bundles with the structure group the projective unitary group of a complex Hilbert space, over a topological space $H$ is given by the third cohomology $text{H}^3(H, Bbb Z)$. When $H$ is a topological group the integral cohomology is often related to a locally continuous (or in the case of a Lie group, locally smooth) third group cohomology of $H$. We shall study in more detail this relation in the case of a group extension $1to N to G to H to 1$ when the gerbe is defined by an abelian extension $1to A to hat N to N to 1$ of $N$. In particular, when $text{H}_s^1(N,A)$ vanishes we shall construct a transgression map $text{H}^2_s(N, A) to text{H}^3_s(H, A^N)$, where $A^N$ is the subgroup of $N$-invariants in $A$ and the subscript $s$ denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper.
We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $mathfrak g$ of left-invariant vector fields on a Lie group $mathbb G$ and we assume that $S$ Lie generates $mathfrak g$. We say that a function $f:mathbb Gto mathbb R$ (or more generally a distribution on $mathbb G$) is $S$-polynomial if for all $Xin S$ there exists $kin mathbb N$ such that the iterated derivative $X^k f$ is zero in the sense of distributions. First, we show that all $S$-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent $k$ in the previous definition is independent on $Xin S$, they form a finite-dimensional vector space. Second, if $mathbb G$ is connected and nilpotent we show that $S$-polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of $mathfrak g$ are equivalent notions.
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