Cohomology of Lie algebras

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A special case of cohomology of algebras. Let be a Lie algebra over a commutative ring with an identity, and suppose that a left -module has been given, that is, a -linear representation of in the -module . The -dimensional cohomology module of the Lie algebra with values in the module is the module , where is the universal enveloping algebra of [3]. In other words, the correspondence is the -th right derived functor of the functor from the category of -modules into the category of -modules, where . The functor is a cohomology functor (see Homology functor).

In small dimensions, the cohomology of Lie algebras can be interpreted as follows. The module is just . If and are -modules, then can be identified with the set of equivalence classes of extensions of the -module with kernel . If is considered as a -module with respect to the adjoint representation (cf. Adjoint representation of a Lie group), then is isomorphic to the quotient module of the module of all derivations (cf. Derivation in a ring) by the submodule of inner derivations. If is a free -module (for example, if is a field), then can be identified with the set of equivalence classes of extensions of the kernels of which are the Abelian Lie algebra with the given representation of . The module can be interpreted also as the set of infinitesimal deformations of the Lie algebra (cf. Deformation).

The following relation exists between the cohomology of Lie algebras and the cohomology of associative algebras; if is a free -module and is an arbitrary two-sided -module, then , where the representation of the algebra in is defined via the formula .

Another way of defining the cohomology of Lie algebras (see [6], [14]) is by using the cochain complex , where is the module of all skew-symmetric -linear mappings , equipped with the coboundary acting by

where the symbol means that the relevant argument is deleted. If is a free -module, the cohomology modules of this complex are naturally isomorphic to the modules . To every subalgebra is associated a subcomplex , leading to the relative cohomology . If is an algebra over on which acts by derivations, then a natural multiplication arises in the cohomology modules, turning into a graded algebra.

Let be the Lie algebra (over ) of smooth vector fields on a differentiable manifold , and let be the space of smooth functions on with the natural -module structure. The definition of the coboundary in coincides formally with that of exterior differentiation of a differential form. More exactly, the de Rham complex (cf. Differential form) is the subcomplex of consisting of the cochains that are linear over . On the other hand, if is the Lie algebra of a connected real Lie group , then the complex can be identified with the complex of left-invariant differential forms on . Analogously, if is the subalgebra corresponding to a connected closed subgroup , then is naturally isomorphic to the complex of -invariant differential forms on the manifold . In particular, if is compact, there follow the isomorphisms of graded algebras:

Precisely these facts serve as starting-point for the definition of cohomology of Lie algebras. Based on them also is the application of the apparatus of the cohomology theory of Lie algebras to the study of the cohomology of principal bundles and homogeneous spaces (see [8], [14]).

The homology of a Lie algebra with coefficients in a right -module is defined in the dual manner. The -dimensional homology group is the -module . In particular, , and if is a trivial -module, .

In calculating the cohomology of a Lie algebra, the following spectral sequences are extensively used; they are often called the Hochschild–Serre spectral sequences. Let be an ideal of and let be a -module. If and are free -modules, there exists a spectral sequence , with , converging to (see [3], [14]). Similar spectral sequences exist for the homology [3]. Further, let be a finite-dimensional Lie algebra over a field of characteristic 0, let be subalgebras such that is reductive in (cf. Lie algebra, reductive), and let be a semi-simple -module. Then there exists a spectral sequence , with , converging to (see [12], [14]).

The cohomology of finite-dimensional reductive (in particular, semi-simple) Lie algebras over a field of characteristic 0 has been investigated completely. If is a finite-dimensional semi-simple Lie algebra over such a field, the following results hold for every finite-dimensional -module :

(Whitehead's lemma). The first of these properties is a sufficient condition for the semi-simplicity of a finite-dimensional algebra , and is equivalent to the semi-simplicity of all finite-dimensional -modules. The second property is equivalent to Levi's theorem (see Levi–Mal'tsev decomposition) for Lie algebras with an Abelian radical [1], [5], [14]. If is a reductive Lie algebra, is a subalgebra of it and is a finite-dimensional semi-simple module, then , which reduces the calculation of the cohomology to the case of the trivial -module (see [5], [14]). The cohomology algebra of a reductive Lie algebra is naturally isomorphic to the algebra of cochains invariant under . In this case is a Hopf algebra, and thus is an exterior algebra over the space of primitive elements, graded in odd degrees , . In particular, is the dimension of the centre of , and is isomorphic to the space of invariant quadratic forms on (see [12], [14]). If is algebraically closed, then is the rank of the algebra , that is, the dimension of its Cartan subalgebra , and are the degrees of the free generators in the algebra of polynomials over invariant under (or in the algebra of polynomials over invariant under the Weyl group, which is isomorphic to it). In this case the numbers are the dimensions of the primitive cohomology classes of the corresponding compact Lie group. The numbers are called the exponents of the Lie algebra . The homology algebra of a reductive Lie algebra over a field of characteristic 0 is the exterior algebra dual to . For any -dimensional Lie algebra , an analogue of Poincaré duality holds:

where and is an arbitrary -dimensional reductive subalgebra of (see [14], ).

Only a few general assertions are known about the cohomology of solvable Lie algebras. For example, let be a finite-dimensional nilpotent Lie algebra over an infinite field and let be a finite-dimensional -module. Then for all if has no trivial -submodules, and for , and for if such a -submodule does exist (see [7]). The groups , are well-studied in the case that is the nilpotent radical of the parabolic subalgebra of some semi-simple Lie algebra over an algebraically closed field of characteristic 0, and the representation of in is the restriction of some representation of in (see [11]). These cohomology groups are closely related to those of the complex homogeneous space corresponding to the pair , with values in sheaves of germs of holomorphic sections of homogeneous vector bundles over . In the calculation of the cohomology of a finite-dimensional non-semi-simple Lie algebra over a field of characteristic 0, one uses the formula

where is an ideal in such that is semi-simple [14].

In some cases, a relation can be established between the cohomology of Lie algebras and the cohomology of groups. Let be a connected real Lie group, let be a maximal compact subgroup of it, let be their Lie algebras, and let be a finite-dimensional smooth -module. If a natural -module structure is defined on , then is isomorphic to the cohomology of (as an abstract group), calculated by means of continuous cochains [10]. On the other hand, let be the Lie algebra of a simply-connected solvable Lie group , let be a lattice in and let be a smooth finite-dimensional linear representation. If is Zariski dense in the algebraic closure of , then (see [4]). In general, . For nilpotent it suffices to require that be unipotent. If the lattice in a simply-connected Lie group is such that is dense in the algebraic closure of the group (for example, if is nilpotent), then .

In recent years there has been a systematic study of the cohomology of certain infinite-dimensional Lie algebras. Among these are the algebra of vector fields on a differentiable manifold , the Lie algebra of formal vector fields, the subalgebras of these algebras consisting of the gradient-free, Hamiltonian or canonical vector fields (see [2], [13]), and also certain classical Banach Lie algebras.

References

 [1] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) [2] D.B. [D.B. Fuks] Fuchs, "Cohomology of infinite-dimensional Lie algebras" , Plenum (1986) (Translated from Russian) [3] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) [4] M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972) [5] , Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Ecole Norm. Sup. (1955) [6] C. Chevalley, S. Eilenberg, "Cohomology theory of Lie groups and Lie algebras" Trans. Amer. Math. Soc. , 63 (1948) pp. 85–124 [7] J. Dixmier, "Cohomologie des algèbres de Lie nilpotents" Acta Sci. Mat. Szeged , 16 : 3–4 (1955) pp. 246–250 [8] W. Greub, S. Halperin, R. Vanstone, "Connections, curvature and cohomology. Cohomology of principal bundles and homogeneous spaces" , 3 , Acad. Press (1975) [9] P. de la Harpe, "Classical Banach–Lie algebras and Banach–Lie groups of operators in Hilbert space" , Springer (1972) [10] G. Hochschild, G.D. Mostow, "Cohomology of Lie groups" Ill. J. Math. , 6 : 3 (1962) pp. 367–401 [11] B. Kostant, "Lie algebra cohomology and the generalized Borel–Weil theorem" Ann. Math. , 74 : 2 (1961) pp. 329–387 [12] J.L. Koszul, "Homologie et cohomologie des algèbres de Lie" Bull. Soc. Math. France , 78 (1950) pp. 65–127 [13] A. Lichnerowicz, "Cohomologie 1-différentiables des algèbres de Lie attaché à une variété symplectique ou de contact" J. Math. Pures Appl. , 53 : 4 (1974) pp. 459–483 [14] A. Verona, "Introducere in coomologia algebrelor Lie" , Bucharest (1974) [15] A. Guichardet, "Cohomologie des groupes topologiques et des algèbres de Lie" , F. Nathan (1980)