nLab Lie group cohomology

Idea

Lie group cohomology generalizes the notion of group cohomology from discrete groups to Lie groups.

From the nPOV on cohomology, a natural definition is that for $G$ a Lie group, its cohomology is the intrinsic cohomology of its delooping Lie groupoid $\mathbf{B}G$ in the (∞,1)-topos $\mathbf{H} =$ Smth$\infty$Grpd.

In the literature one finds a sequence of definitions that approach this intrisic topos-theoretic definition. This is discussed below. For a detailed discussion of the relation of this to the intrinsic topos-theoretic definition see the section Cohomology of Lie groups at ∞-Lie groupoid.

Structured group cohomology

If the groups in question are not group objects internal to Set but groups with extra structure, such as topological groups or Lie groups, then their cohomology has to be understood in the corresponding natural context.

In parts of the literature cohomology of structured groups $G$ is defined in direct generalization of the formulas above as homotopy classes of morphisms from the simplicial object

$$\left( \cdots G \times G\stackrel{\to}{\stackrel{\to}{\to}}G \stackrel{\to}{\to} * \right)$$

to a simplicial object $N (\mathbf{B}^n A)$.

This is what is described above. But this does not in general give the right answer for structured groups:

namely cohomology is really about homotopy classes of maps in the suitable ambient (∞,1)-topos. For plain groups as in the above entry, we are working in the $(\infty,1)$-topos ∞Grpd. That may be modeled by the standard model structure on simplicial sets. In that model structure, all objects a cofibrant and Kan complexes are fibrant. That means all objects we are dealing with here are both cofibrant and fibrant, and hence the simplicial set of maps between them is the correct derived hom-space between these objects.

But this changes as we consider groups with extra structure. For a Lie group $G$, the object

$$\left( \cdots G \times G\stackrel{\to}{\stackrel{\to}{\to}}G \stackrel{\to}{\to} * \right)$$

has to be considered as an Lie ∞-groupoid: an object in the model structure on simplicial presheaves over a site such as Diff or CartSp. As such it is in general not both cofibrant and fibrant. To that extent plain morphisms out of this object do not compute the correct derived hom-spaces. Instead, the right definition of structured group cohomology uses the correct fibrant and cofibrant replacements.

Topological group cohomology

In

• Jim Stasheff, Continuous cohomology of groups and classifying spaces Bull. Amer. Math. Soc. Volume 84, Number 4 (1978), 513-530 (web)

$n$-cocycles on a topological group $G$ with valzues in a topological abelian group $A$ are considered as continuous functions $G^{\times n}\to A$ (p. 3 ). (“continuous cohomology”)

A definition in terms of Ext-functors and comparison with the naive definition is in

• David Wigner, Algebraic cohomology of topological groups Transactions of the American Mathematical Society, volume 178 (1973)(pdf)

A classical reference that considers the cohomology of Lie groups as topological spaces is

• Armand Borel, Homology and cohomology of compact connected Lie groups (pdf)

A corrected definition of topological group cohomology has been given by Segal

• Graeme Segal, Cohomology of topological groups In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377{387. Academic Press, London, (1970).

• Graeme Segal, A classifying space of a topological group in the sense of Gel’fand-Fuks. Funkcional. Anal. i Prilozen., 9(2):48{50, (1975).

Definition

For $G$ a Lie group and $A$ an abelian Lie group, write

$$H_{naive}^n(G,A) = \{smooth G^{\times n } \to A\}/_\sim$$

for the naive notion of cohomology on $G$.

A refined definition of Lie group cohomology, denoted $H^n_{diff}(G,A)$, was given in (Brylinski) following (Blanc) and effectively rediscovers Segal’s definition. See section 4 of (Schommer-Pries) for a review and applications.

Definition

(Brylinski)

Let $G$ be a Lie group (paracompact) and $A$ an abelian Lie group.

For eack $k \in \mathbb{N}$ we can pick a good open cover $\{U^{k}_{i} \to G^{\times_k}| i \in I_k\}$ such that

• the index sets arrange themselves into a simplicial set $I : [k] \mapsto I_k$;

• and for $d_j(U^k_i)$ and $s_j(U^k_i)$ the images of the face and degeneracy maps of $G^{\times\bullet}$ we have

  $$d_j(U^k_i) \subset U^{k-1}_{d_j(i)}$$

  and

  $$s_j(U^k_i) \subset U^{k+1}_{s_j(i)} \,.$$

Then the differentiable group cohomology of $G$ with coefficients in $A$ is the cohomology of the total complex of the Cech double complex $C^\infty( U^{\bullet}_{i_0, \cdots, i_\bullet} , A)$ whose differentials are the alternating sums of the face maps of $G^{\times_\bullet}$ and of the Cech nerves, respectively:

$$H^n_{diff}(G,A) := H^n Tot C^\infty( U^{\bullet}_{i_0, \cdots, i_\bullet} , A)$$

This is definition 1.1 in (Brylinski)

As discussed there, this is equivalent to other definitions, notably to a definition given earlier by Graeme Segal.

There is an evident morphism

$$H^n_{naive}(G,A) \to H^n_{diff}(G,A)$$

obtained by pulling back a globally defined smooth cocycle to a cover.

At ∞-Lie groupoid it is discussed that there is a further refinement

$$H^n_{diff}(G,A) \to H^n(\mathbf{B}G,A) \,,$$

where on the right we have the intrinsic cohomology of ∞-Lie groupoids.

Properties

General

From now on, for definiteness by a Lie group $G$ we mean (following Bry, page3) a paracompact Frechet manifold equipped with a group structure such that the product and the inverse maps are smooth, and there is an everywhere defined exponential map $exp : \mathfrak{g} \to G$ where $\mathfrak{g}$ is the Lie algebra of $G$.

Proposition

If the coefficient Lie group $A$ is a topological vector space, then the naive group cohomology $H^n(G,A) = \{smooth G^{\times n} \to A\}/_\sim$ coincides with the correct Lie group cohomology

$$(A top.\;vect.\;space) \Rightarrow (H^n_{naive}(G,A) \stackrel{\simeq}{\to} H^n_{diff}(G,A) ) \,.$$

If the coefficient Lie group $A$ is discrete, then Lie group cohomology coindices with the topological cohomology of the classifying space $\mathcal{B}G$

$$(A discrete) \Rightarrow (H^n(G,A) \simeq H^n(\mathcal{B}G), A) \,.$$

Proof

This is Bry, prop. 1.3 and Bry, lemma 1.5.

Proposition

$H^2_{diff}(G,A)$ classifies central extensions of Lie groups

$$A \to \hat G \stackrel{\pi}{\to} G$$

such that $\pi : \hat G \to G$ is a locally trivial smooth principal $A$-fibration.

The image of $H^2_{naive}(G,A) \to H^2_{diff}(G,A)$ consists of those central extensions for which is bundle is trivial.

Proof

This is Bry, prop. 1.6 and Bry, lemma 1.5.

Long sequence in differential cohomology

For the purpose of this section we specifically conceive Lie group cohomology inside the (∞,1)-topos ∞LieGrpd of ∞-Lie groupoids, as described there.

This is a local (∞,1)-topos, hence in particular an ∞-connected (∞,1)-topos and therefore it admits differential cohomology in an (∞,1)-topos. By the theorem about the differential fiber sequence we have for $G$ a Lie group, $\mathbf{B}G$ its delooping, $\mathbf{B}^{n} U(1)$ the circle n+1-group a long sequence in cohomology

$$\cdots \to H^n_{diff}(G, (1))\to H^n_G(G,U(1)) \to H_{dR, G}^{n+1}(G) \to \cdots \,,$$

where $H_G(G,-)$ denotes $G$-equivariant cohomology, in that

$$H^n_G(G,A) := \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A)$$

and so on.

The point to note is that we may identify Lie algebra cohomology inside $H^n_{dR,G}(G)$ and may therefore regard the map

$$H^n_G(G,U(1)) \to H_{dR, G}^{n+1}(G)$$

as the differentiation map that take a smooth group cocycle to a Lie algebra cocycle. This morphism operates by putting constructing a circle n-bundle with connection over $\mathbf{B}G$ and then computing its curvature forms.

Example

For $\mathfrak{g}$ a semisimple Lie algebra,$G$ its simply connected Lie group, let $\mathbf{B}G \to \mathbf{B}^3 U(1)$ be the group cocycle that classifies the string 2-group. Its image in $H^3_{dR,G}(G)$ is the curvature of the Chern-Simons circle 3-bundle over $\mathbf{B}G$. This is represented by a simplicial differential form consisting of two pieces

• on $G$ the form $\langle \theta\wedge \theta \wedge \theta \rangle$ obtained by feeding the Maurer-Cartan form on $G$ into the canoical Lie algebra cocycle that is in transtression with the Killing form invariant polynomial;

• on $G \times G$ something like $\langle \theta_1 \wedge \theta_2\rangle$.

(…)

Relation to intrinsic cohomology of smooth $\infty$-groupoids

We may naturally regard a Lie group as an ∞-group in the cohesive (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids. As such, there is an intrinsic (∞,1)-topos-theoretic notion of its cohomology.

Proposition

For

1. $G$ a Lie group and $A$ either a discrete group

2. $G$ a compact Lie group and $A$ the additive Lie group of real numbers $\mathbb{R}$ or the circle group $\mathbb{R}/Z = U(1)$

the intrinsic cohomology of $G$ in Smooth∞Grpd coincides with the refined Lie group cohomology of (Segal)(Brylinski)

$$H^n_{Smooth\infty Grpd}(\mathbf{B}G, A) \simeq H^n_{diffr}(G,A) \,.$$

This is discussed in detail at Smooth∞Grpd and proven at SynthDiff∞Grpd.

Relation to Lie algebra cohomology

The content of a van Est isomorphism is that the canonical comparison map from Lie group cohomology to Lie algebra cohomology (by differentiation) is an isomorphism whenever the Lie group is sufficiently connected.

Related entries

• group cohomology

  • nonabelian group cohomology, groupoid cohomology

  • ∞-group cohomology

• group extension

• Lie group cohomology

  • ∞-Lie groupoid cohomology

References

A textbook account of standard material is in chapter V in vol III of

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)

The definition of the refined topological group cohomology in terms of degreewise abelian sheaf cohomology was given in

• Graeme Segal, Cohomology of topological groups,

  Symposia mathematica. Volume IV, pp. 377–387. Convegni del Dicembre del 1968 e del Marzo del 1969. Istituto Nazionale di Alta Matematica, Roma. Academic Press, London–New York, 1970. iv+542 pp. PDF.

It was later rediscovered for Lie groups in

• Jean-Luc Brylinski, Differentiable Cohomology of Gauge Groups (arXiv)

following

• P. Blanc, Cohomologie différentiable et changement de groupes, Astérisque vol. 124-125 (1985), pp. 113-130

Further discussion:

• Friedrich Wagemann, Christoph Wockel, A Cocycle Model for Topological and Lie Group Cohomology, Transactions of the American Mathematical Society 367 3 (2015) 1871-1909 (arXiv:1110.3304, jstor:24513021)

Relevant background on the theory of abelian sheaf cohomology on simplicial spaces is at the beginning of

• Pierre Deligne, Théorie de Hodge: III Publication mathétematique de l’I.H.E.S, tome 44 (1974), p. 5-77 (numdam)

More discussion of Lie group cohomology along these lines is in

• Chris Schommer-Pries, A finite-dimensional String 2-group (arXiv:0911.2483)

A discussion of the relation between local Lie group cohomology and Lie algebra cohomology is in

• S. Świerczkowski, Cohomology of group germs and Lie algebras Pacific Journal of Mathematics, Volume 39, Number 2 (1971), 471-482. (pdf)