View source for Cohomology of Lie algebras

A special case of cohomology of algebras. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c0231401.png" /> be a Lie algebra over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c0231402.png" /> with an identity, and suppose that a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c0231403.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c0231404.png" /> has been given, that is, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c0231405.png" />-linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c0231406.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c0231407.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c0231408.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314010.png" />-dimensional cohomology module of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314011.png" /> with values in the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314012.png" /> is the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314014.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314015.png" /> is the universal enveloping algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314016.png" /> [[#References|[3]]]. In other words, the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314017.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314018.png" />-th right [[Derived functor|derived functor]] of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314019.png" /> from the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314020.png" />-modules into the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314021.png" />-modules, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314022.png" />. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314023.png" /> is a cohomology functor (see [[Homology functor|Homology functor]]).

In small dimensions, the cohomology of Lie algebras can be interpreted as follows. The module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314024.png" /> is just <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314027.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314028.png" />-modules, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314029.png" /> can be identified with the set of equivalence classes of extensions of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314030.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314031.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314032.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314033.png" /> is considered as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314034.png" />-module with respect to the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314035.png" /> (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314036.png" /> is isomorphic to the quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314037.png" /> of the module of all derivations (cf. [[Derivation in a ring|Derivation in a ring]]) by the submodule of inner derivations. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314038.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314039.png" />-module (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314040.png" /> is a field), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314041.png" /> can be identified with the set of equivalence classes of extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314042.png" /> the kernels of which are the Abelian Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314043.png" /> with the given representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314044.png" />. The module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314045.png" /> can be interpreted also as the set of infinitesimal deformations of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314046.png" /> (cf. [[Deformation|Deformation]]).

The following relation exists between the cohomology of Lie algebras and the cohomology of associative algebras; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314047.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314048.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314049.png" /> is an arbitrary two-sided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314050.png" />-module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314051.png" />, where the representation of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314052.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314053.png" /> is defined via the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314054.png" />.

Another way of defining the cohomology of Lie algebras (see [[#References|[6]]], [[#References|[14]]]) is by using the cochain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314056.png" /> is the module of all skew-symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314057.png" />-linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314058.png" />, equipped with the coboundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314059.png" /> acting by

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where the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314064.png" /> means that the relevant argument is deleted. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314065.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314066.png" />-module, the cohomology modules of this complex are naturally isomorphic to the modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314067.png" />. To every subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314068.png" /> is associated a subcomplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314069.png" />, leading to the relative cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314070.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314071.png" /> is an algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314072.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314073.png" /> acts by derivations, then a natural multiplication arises in the cohomology modules, turning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314074.png" /> into a graded algebra.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314075.png" /> be the Lie algebra (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314076.png" />) of smooth vector fields on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314077.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314078.png" /> be the space of smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314079.png" /> with the natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314080.png" />-module structure. The definition of the coboundary in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314081.png" /> coincides formally with that of exterior differentiation of a differential form. More exactly, the de Rham complex (cf. [[Differential form|Differential form]]) is the subcomplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314082.png" /> consisting of the cochains that are linear over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314083.png" />. On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314084.png" /> is the Lie algebra of a connected real Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314085.png" />, then the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314086.png" /> can be identified with the complex of left-invariant differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314087.png" />. Analogously, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314088.png" /> is the subalgebra corresponding to a connected closed subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314089.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314090.png" /> is naturally isomorphic to the complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314091.png" />-invariant differential forms on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314092.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314093.png" /> is compact, there follow the isomorphisms of graded algebras:

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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 [[#References|[8]]], [[#References|[14]]]).

The homology of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314095.png" /> with coefficients in a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314096.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314097.png" /> is defined in the dual manner. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314098.png" />-dimensional homology group is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c02314099.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140100.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140101.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140102.png" /> is a trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140103.png" />-module, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140104.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140105.png" /> be an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140106.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140107.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140108.png" />-module. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140110.png" /> are free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140111.png" />-modules, there exists a [[Spectral sequence|spectral sequence]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140112.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140113.png" />, converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140114.png" /> (see [[#References|[3]]], [[#References|[14]]]). Similar spectral sequences exist for the homology [[#References|[3]]]. Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140115.png" /> be a finite-dimensional Lie algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140116.png" /> of characteristic 0, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140117.png" /> be subalgebras such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140118.png" /> is reductive in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140119.png" /> (cf. [[Lie algebra, reductive|Lie algebra, reductive]]), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140120.png" /> be a semi-simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140121.png" />-module. Then there exists a spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140122.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140123.png" />, converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140124.png" /> (see [[#References|[12]]], [[#References|[14]]]).

The cohomology of finite-dimensional reductive (in particular, semi-simple) Lie algebras over a field of characteristic 0 has been investigated completely. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140125.png" /> is a finite-dimensional semi-simple Lie algebra over such a field, the following results hold for every finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140126.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140127.png" />:

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(Whitehead's lemma). The first of these properties is a sufficient condition for the semi-simplicity of a finite-dimensional algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140129.png" />, and is equivalent to the semi-simplicity of all finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140130.png" />-modules. The second property is equivalent to Levi's theorem (see [[Levi–Mal'tsev decomposition|Levi–Mal'tsev decomposition]]) for Lie algebras with an Abelian radical [[#References|[1]]], [[#References|[5]]], [[#References|[14]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140131.png" /> is a reductive Lie algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140132.png" /> is a subalgebra of it and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140133.png" /> is a finite-dimensional semi-simple module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140134.png" />, which reduces the calculation of the cohomology to the case of the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140135.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140136.png" /> (see [[#References|[5]]], [[#References|[14]]]). The cohomology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140137.png" /> of a reductive Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140138.png" /> is naturally isomorphic to the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140139.png" /> of cochains invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140140.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140141.png" /> is a Hopf algebra, and thus is an exterior algebra over the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140142.png" /> of primitive elements, graded in odd degrees <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140144.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140145.png" /> is the dimension of the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140146.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140147.png" /> is isomorphic to the space of invariant quadratic forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140148.png" /> (see [[#References|[12]]], [[#References|[14]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140149.png" /> is algebraically closed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140150.png" /> is the rank of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140151.png" />, that is, the dimension of its Cartan subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140152.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140153.png" /> are the degrees of the free generators in the algebra of polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140154.png" /> invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140155.png" /> (or in the algebra of polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140156.png" /> invariant under the Weyl group, which is isomorphic to it). In this case the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140157.png" /> are the dimensions of the primitive cohomology classes of the corresponding compact Lie group. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140158.png" /> are called the exponents of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140159.png" />. The homology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140160.png" /> of a reductive Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140161.png" /> over a field of characteristic 0 is the exterior algebra dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140162.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140163.png" />-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140164.png" />, an analogue of Poincaré duality holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140165.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140166.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140167.png" /> is an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140168.png" />-dimensional reductive subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140169.png" /> (see [[#References|[14]]], ).

Only a few general assertions are known about the cohomology of solvable Lie algebras. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140170.png" /> be a finite-dimensional nilpotent Lie algebra over an infinite field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140171.png" /> be a finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140172.png" />-module. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140173.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140174.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140175.png" /> has no trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140176.png" />-submodules, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140177.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140178.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140179.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140180.png" /> if such a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140181.png" />-submodule does exist (see [[#References|[7]]]). The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140182.png" />, are well-studied in the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140183.png" /> is the nilpotent radical of the parabolic subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140184.png" /> of some semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140185.png" /> over an algebraically closed field of characteristic 0, and the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140186.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140187.png" /> is the restriction of some representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140188.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140189.png" /> (see [[#References|[11]]]). These cohomology groups are closely related to those of the complex homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140190.png" /> corresponding to the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140191.png" />, with values in sheaves of germs of holomorphic sections of homogeneous vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140192.png" />. 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140193.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140194.png" /> is an ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140195.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140196.png" /> is semi-simple [[#References|[14]]].

In some cases, a relation can be established between the cohomology of Lie algebras and the [[Cohomology of groups|cohomology of groups]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140197.png" /> be a connected real Lie group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140198.png" /> be a maximal compact subgroup of it, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140199.png" /> be their Lie algebras, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140200.png" /> be a finite-dimensional smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140201.png" />-module. If a natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140202.png" />-module structure is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140203.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140204.png" /> is isomorphic to the cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140205.png" /> (as an abstract group), calculated by means of continuous cochains [[#References|[10]]]. On the other hand, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140206.png" /> be the Lie algebra of a simply-connected solvable Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140207.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140208.png" /> be a lattice in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140209.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140210.png" /> be a smooth finite-dimensional linear representation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140211.png" /> is Zariski dense in the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140212.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140213.png" /> (see [[#References|[4]]]). In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140214.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140215.png" />. For nilpotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140216.png" /> it suffices to require that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140217.png" /> be unipotent. If the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140218.png" /> in a simply-connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140219.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140220.png" /> is dense in the algebraic closure of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140221.png" /> (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140222.png" /> is nilpotent), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140223.png" />.

In recent years there has been a systematic study of the cohomology of certain infinite-dimensional Lie algebras. Among these are the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140224.png" /> of vector fields on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140225.png" />, the Lie algebra of formal vector fields, the subalgebras of these algebras consisting of the gradient-free, Hamiltonian or canonical vector fields (see [[#References|[2]]], [[#References|[13]]]), and also certain classical Banach Lie algebras.

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



====Comments====
The subcomplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140226.png" /> of relative cochains is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140227.png" />. Equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140228.png" />.

There is a generalization of the Poincaré duality result as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140229.png" /> be free of finite dimension over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140230.png" />. For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140231.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140232.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140233.png" /> be the dual Lie module defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140234.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140235.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140236.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140237.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140238.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140239.png" />-module with underlying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140240.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140241.png" /> but with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140242.png" />-action changed to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140243.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140244.png" /> is the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140245.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140246.png" />. Then there is a canonical isomorphism, [[#References|[a1]]],

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140247.png" /></td> </tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140248.png" />-modules where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140249.png" />. Note that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140250.png" /> is semi-simple then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023140/c023140251.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hazewinkel,   "A duality theorem for the cohomology of Lie algebras"  ''Math. USSR-Sb.'' , '''12'''  (1970)  pp. 638–644  ''Mat. Sb.'' , '''83 (125)'''  (1970)  pp. 639–644</TD></TR></table>