Cohomology of Lie algebras

A special case of cohomology of algebras. Let $ \mathfrak G $ be a Lie algebra over a commutative ring $ K $ with an identity, and suppose that a left $ \mathfrak G $- module $ V $ has been given, that is, a $ K $- linear representation of $ \mathfrak G $ in the $ K $- module $ V $. The $ p $- dimensional cohomology module of the Lie algebra $ \mathfrak G $ with values in the module $ V $ is the module $ H ^ {p} ( \mathfrak G , V) = \mathop{\rm Ext} _ {U \mathfrak G } ^ {p} ( K, V) $, $ p = 0, 1 \dots $ where $ U \mathfrak G $ is the universal enveloping algebra of $ \mathfrak G $[3]. In other words, the correspondence $ V \mapsto H ^ {p} ( \mathfrak G , V) $ is the $ p $- th right derived functor of the functor $ V \mapsto V ^ {\mathfrak G} $ from the category of $ \mathfrak G $- modules into the category of $ K $- modules, where $ V ^ {\mathfrak G} = \{ {v \in V } : {xv = 0 ( x \in \mathfrak G ) } \} $. The functor $ V \mapsto H ^ {*} ( \mathfrak G , V) = \sum _ {p \geq 0 } H ^ {p} ( \mathfrak G , V) $ is a cohomology functor (see Homology functor).

In small dimensions, the cohomology of Lie algebras can be interpreted as follows. The module $ H ^ {0} ( \mathfrak G , V) $ is just $ V ^ {\mathfrak G} $. If $ V ^ { \prime } $ and $ V ^ { \prime\prime } $ are $ \mathfrak G $- modules, then $ H ^ {1} ( \mathfrak G , \mathop{\rm Hom} _ {K} ( V ^ { \prime\prime } , V ^ { \prime } )) $ can be identified with the set of equivalence classes of extensions of the $ \mathfrak G $- module $ V ^ { \prime\prime } $ with kernel $ V ^ { \prime } $. If $ \mathfrak G $ is considered as a $ \mathfrak G $- module with respect to the adjoint representation $ \mathop{\rm ad} $( cf. Adjoint representation of a Lie group), then $ H ^ {1} ( \mathfrak G , \mathfrak G ) $ is isomorphic to the quotient module $ \mathop{\rm Der} \mathfrak G / \mathop{\rm ad} \mathfrak G $ of the module of all derivations (cf. Derivation in a ring) by the submodule of inner derivations. If $ \mathfrak G $ is a free $ K $- module (for example, if $ K $ is a field), then $ H ^ {2} ( \mathfrak G , V) $ can be identified with the set of equivalence classes of extensions of $ \mathfrak G $ the kernels of which are the Abelian Lie algebra $ V $ with the given representation of $ \mathfrak G $. The module $ H ^ {2} ( \mathfrak G , \mathfrak G ) $ can be interpreted also as the set of infinitesimal deformations of the Lie algebra $ \mathfrak G $( cf. Deformation).

The following relation exists between the cohomology of Lie algebras and the cohomology of associative algebras; if $ \mathfrak G $ is a free $ K $- module and $ V $ is an arbitrary two-sided $ U \mathfrak G $- module, then $ H ^ {p} ( U \mathfrak G , V) \cong H ^ {p} ( \mathfrak G , V) $, where the representation of the algebra $ \mathfrak G $ in $ V $ is defined via the formula $ ( x, v) \mapsto xv - vx $.

Another way of defining the cohomology of Lie algebras (see [6], [14]) is by using the cochain complex $ C ^ {*} ( \mathfrak G , V) = \sum _ {p \geq 0 } C ^ {p} ( \mathfrak G , V) $, where $ C ^ {p} ( \mathfrak G , V) = C ^ {p} $ is the module of all skew-symmetric $ p $- linear mappings $ \mathfrak G ^ {p} \rightarrow V $, equipped with the coboundary $ d: C ^ {p} \rightarrow C ^ {p + 1 } $ acting by

$$ ( d \omega ) ( x _ {1} \dots x _ {p + 1 } ) = $$

$$ = \ \sum _ {i = 1 } ^ { {p } + 1 } (- 1) ^ {i + 1 } x _ {i} \omega ( x _ {1} \dots \widehat{x} _ {i} \dots x _ {p + 1 } ) + $$

$$ + \sum _ {i < j } (- 1) ^ {i + j } \omega ([ x _ {i} , x _ {j} ], x _ {1} \dots \widehat{x} _ {i} \dots \widehat{x} _ {j} \dots x _ {p + 1 } ) , $$

$$ \omega \in C ^ {p} ; \ x _ {1} \dots x _ {p + 1 } \in \mathfrak G , $$

where the symbol $ \widehat{ {}} $ means that the relevant argument is deleted. If $ G $ is a free $ K $- module, the cohomology modules of this complex are naturally isomorphic to the modules $ H ^ {p} ( \mathfrak G , V) $. To every subalgebra $ \mathfrak H \subset \mathfrak G $ is associated a subcomplex $ C ^ {*} ( \mathfrak G , \mathfrak H ; V) \subset C ^ {*} ( \mathfrak G , V) $, leading to the relative cohomology $ H ^ {*} ( \mathfrak G , \mathfrak H ; V) = \sum _ {p \geq 0 } H ^ {p} ( \mathfrak G , \mathfrak H ; V) $. If $ V $ is an algebra over $ K $ on which $ \mathfrak G $ acts by derivations, then a natural multiplication arises in the cohomology modules, turning $ H ^ {*} ( \mathfrak G , \mathfrak H , V) $ into a graded algebra.

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

$$ H ^ {*} ( \mathfrak G , \mathbf R ) \cong \ H ^ {*} ( G, \mathbf R ); \ \ H ^ {*} ( \mathfrak G , \mathfrak H ; \mathbf R ) \cong \ H ^ {*} ( G/H, \mathbf R ). $$

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 $ \mathfrak G $ with coefficients in a right $ \mathfrak G $- module $ V $ is defined in the dual manner. The $ p $- dimensional homology group is the $ K $- module $ H _ {p} ( \mathfrak G , V) = \mathop{\rm Tor} _ {p} ^ {U \mathfrak G } ( V, K) $. In particular, $ H _ {0} ( \mathfrak G , V) = V/V \mathfrak G $, and if $ V $ is a trivial $ \mathfrak G $- module, $ H _ {1} ( \mathfrak G , V) \cong V \otimes _ {K} \mathfrak G /[ \mathfrak G , \mathfrak G ] $.

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 $ \mathfrak H $ be an ideal of $ \mathfrak G $ and let $ V $ be a $ \mathfrak G $- module. If $ \mathfrak H $ and $ \mathfrak G / \mathfrak H $ are free $ K $- modules, there exists a spectral sequence $ \{ E _ {r} ^ {p,q} \} $, with $ \{ E _ {2} ^ {p,q} \} = H ^ {p} ( \mathfrak G / \mathfrak H , H ^ {q} ( \mathfrak H , V)) $, converging to $ H ^ {*} ( \mathfrak G , V) $( see [3], [14]). Similar spectral sequences exist for the homology [3]. Further, let $ \mathfrak G $ be a finite-dimensional Lie algebra over a field $ K $ of characteristic 0, let $ \mathfrak G ^ { \prime\prime } \subset \mathfrak G ^ { \prime } $ be subalgebras such that $ \mathfrak G ^ { \prime } $ is reductive in $ \mathfrak G $( cf. Lie algebra, reductive), and let $ V $ be a semi-simple $ \mathfrak G $- module. Then there exists a spectral sequence $ \{ F _ {r} ^ { p,q } \} $, with $ F _ {2} ^ { p,q } = H ^ {p} ( \mathfrak G , \mathfrak G ^ { \prime } ; V) \otimes H ^ {q} ( \mathfrak G ^ { \prime } , \mathfrak G ^ { \prime\prime } ; K) $, converging to $ H ^ {*} ( \mathfrak G , \mathfrak G ^ { \prime\prime } ; V) $( 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 $ \mathfrak G $ is a finite-dimensional semi-simple Lie algebra over such a field, the following results hold for every finite-dimensional $ \mathfrak G $- module $ V $:

$$ H ^ {1} ( \mathfrak G , V) = 0; \ \ H ^ {2} ( \mathfrak G , V) = 0 $$

(Whitehead's lemma). The first of these properties is a sufficient condition for the semi-simplicity of a finite-dimensional algebra $ \mathfrak G $, and is equivalent to the semi-simplicity of all finite-dimensional $ \mathfrak G $- 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 $ \mathfrak G $ is a reductive Lie algebra, $ \mathfrak H $ is a subalgebra of it and $ V $ is a finite-dimensional semi-simple module, then $ H ^ {*} ( \mathfrak G , \mathfrak H ; V) \cong H ^ {*} ( \mathfrak G , \mathfrak H ; V ^ {\mathfrak G} ) $, which reduces the calculation of the cohomology to the case of the trivial $ \mathfrak G $- module $ V = K $( see [5], [14]). The cohomology algebra $ H ^ {*} ( \mathfrak G , K) $ of a reductive Lie algebra $ \mathfrak G $ is naturally isomorphic to the algebra $ C ^ {*} ( \mathfrak G , K) ^ {\mathfrak G} $ of cochains invariant under $ \mathop{\rm ad} $. In this case $ H ^ {*} ( \mathfrak G , K) $ is a Hopf algebra, and thus is an exterior algebra over the space $ P _ {\mathfrak G} $ of primitive elements, graded in odd degrees $ 2m _ {i} - 1 $, $ i = 1 \dots r $. In particular, $ \mathop{\rm dim} H ^ {1} ( \mathfrak G , K) = \mathop{\rm dim} P _ {\mathfrak G} ^ {1} $ is the dimension of the centre of $ \mathfrak G $, and $ P _ {\mathfrak G} ^ {3} $ is isomorphic to the space of invariant quadratic forms on $ \mathfrak G $( see [12], [14]). If $ K $ is algebraically closed, then $ r $ is the rank of the algebra $ \mathfrak G $, that is, the dimension of its Cartan subalgebra $ \mathfrak A $, and $ m _ {i} $ are the degrees of the free generators in the algebra of polynomials over $ \mathfrak G $ invariant under $ \mathop{\rm ad} $( or in the algebra of polynomials over $ \mathfrak A $ invariant under the Weyl group, which is isomorphic to it). In this case the numbers $ 2m _ {i} - 1 $ are the dimensions of the primitive cohomology classes of the corresponding compact Lie group. The numbers $ m _ {i} - 1 $ are called the exponents of the Lie algebra $ G $. The homology algebra $ H _ {*} ( \mathfrak G , K) $ of a reductive Lie algebra $ \mathfrak G $ over a field of characteristic 0 is the exterior algebra dual to $ H ^ {*} ( \mathfrak G , K) $. For any $ n $- dimensional Lie algebra $ \mathfrak G $, an analogue of Poincaré duality holds:

$$ H ^ {p} ( \mathfrak G , \mathfrak H ; K) \cong \ H _ {n - m - p } ( \mathfrak G , \mathfrak H ; K), $$

where $ 0 \leq p \leq n - m $ and $ \mathfrak H $ is an arbitrary $ m $- dimensional reductive subalgebra of $ \mathfrak G $( see [14], ).

Only a few general assertions are known about the cohomology of solvable Lie algebras. For example, let $ \mathfrak G $ be a finite-dimensional nilpotent Lie algebra over an infinite field and let $ V $ be a finite-dimensional $ \mathfrak G $- module. Then $ H ^ {p} ( \mathfrak G , V) = 0 $ for all $ p $ if $ V $ has no trivial $ \mathfrak G $- submodules, and $ H ^ {p} ( \mathfrak G , V) \neq 0 $ for $ p = 0 \dots n = \mathop{\rm dim} \mathfrak G $, and $ \mathop{\rm dim} H ^ {p} ( \mathfrak G , K) \geq 2 $ for $ 1 \leq p \leq n - 1 $ if such a $ \mathfrak G $- submodule does exist (see [7]). The groups $ H ^ {p} ( \mathfrak N , V) $, are well-studied in the case that $ \mathfrak N $ is the nilpotent radical of the parabolic subalgebra $ \mathfrak P $ of some semi-simple Lie algebra $ \mathfrak G $ over an algebraically closed field of characteristic 0, and the representation of $ \mathfrak N $ in $ V $ is the restriction of some representation of $ \mathfrak G $ in $ V $( see [11]). These cohomology groups are closely related to those of the complex homogeneous space $ G/P $ corresponding to the pair $ \mathfrak G \supset \mathfrak P $, with values in sheaves of germs of holomorphic sections of homogeneous vector bundles over $ G/P $. 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

$$ H ^ {*} ( \mathfrak G , V) \cong \ H ^ {*} ( \mathfrak G / \mathfrak H , K) \otimes H ^ {*} ( \mathfrak H , V) ^ {\mathfrak G} , $$

where $ \mathfrak H $ is an ideal in $ \mathfrak G $ such that $ \mathfrak G / \mathfrak H $ is semi-simple [14].

In some cases, a relation can be established between the cohomology of Lie algebras and the cohomology of groups. Let $ G $ be a connected real Lie group, let $ K $ be a maximal compact subgroup of it, let $ \mathfrak G \supset \mathfrak K $ be their Lie algebras, and let $ V $ be a finite-dimensional smooth $ G $- module. If a natural $ \mathfrak G $- module structure is defined on $ V $, then $ H ^ {*} ( \mathfrak G , \mathfrak K ; V) $ is isomorphic to the cohomology of $ G $( as an abstract group), calculated by means of continuous cochains [10]. On the other hand, let $ \mathfrak G $ be the Lie algebra of a simply-connected solvable Lie group $ G $, let $ \Gamma $ be a lattice in $ G $ and let $ \rho : G \rightarrow \mathop{\rm GL} ( V) $ be a smooth finite-dimensional linear representation. If $ \rho ( \Gamma ) \times \mathop{\rm Ad} \Gamma $ is Zariski dense in the algebraic closure of $ \rho ( G) \times \mathop{\rm Ad} G $, then $ H ^ {*} ( \mathfrak G , V) \cong H ^ {*} ( \Gamma , V) $( see [4]). In general, $ \mathop{\rm dim} H ^ {p} ( \Gamma , V) \geq \mathop{\rm dim} H ^ {p} ( \mathfrak G , V) $ $ ( p = 0, 1 , . . .) $. For nilpotent $ \mathfrak G $ it suffices to require that $ \rho $ be unipotent. If the lattice $ \Gamma $ in a simply-connected Lie group $ G $ is such that $ \mathop{\rm Ad} \Gamma $ is dense in the algebraic closure of the group $ \mathop{\rm Ad} G $( for example, if $ G $ is nilpotent), then $ H ^ {*} ( \mathfrak G , \mathbf R ) \cong H ( G/ \Gamma , \mathbf R ) $.

In recent years there has been a systematic study of the cohomology of certain infinite-dimensional Lie algebras. Among these are the algebra $ \mathfrak X ( M) $ of vector fields on a differentiable manifold $ M $, 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.

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The subcomplex $ C ^ {*} ( \mathfrak G , \mathfrak H ; V) $ of relative cochains is defined by $ C ^ {q} ( \mathfrak G , \mathfrak H ; V) = \{ {f \in C ^ {q} ( \mathfrak G ; V) } : {f ( g _ {1} \dots g _ {q} ) = 0 \textrm{ and } df ( g _ {1} \dots g _ {q} ) = 0 \textrm{ if } g _ {1} \in \mathfrak H } \} $. Equivalently, $ C ^ {q} ( \mathfrak G , \mathfrak H ; V) = \mathop{\rm Hom} _ {\mathfrak H} ( \wedge ^ {q} ( \mathfrak G / \mathfrak H ) , V) $.

There is a generalization of the Poincaré duality result as follows. Let $ \mathfrak G $ be free of finite dimension over $ K $. For a $ \mathfrak G $- module $ M $ let $ M ^ {*} = \mathop{\rm Hom} ( M, K) $ be the dual Lie module defined by $ \langle uf, m \rangle = \langle f, um \rangle $ for $ f \in M ^ {*} $, $ m \in M $, $ u \in \mathfrak G $, and let $ M ^ {tw} $ be the $ \mathfrak G $- module with underlying $ K $- module $ M $ but with the $ \mathfrak G $- action changed to $ u \mapsto \rho ( u) - \mathop{\rm Tr} ( \mathop{\rm ad} u) \in \mathop{\rm End} _ {K} ( M) $, where $ \rho $ is the action of $ \mathfrak G $ on $ M $. Then there is a canonical isomorphism, [a1],

$$ H ^ {s} ( \mathfrak G, ( M ^ {tw} ) ^ {*} ) \widetilde \rightarrow \ ( H ^ {n - s } ( \mathfrak G , M)) ^ {*} $$

of $ K $- modules where $ n = \mathop{\rm dim} _ {K} \mathfrak G $. Note that if $ \mathfrak G $ is semi-simple then $ M ^ {tw} = M $.

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