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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris Jean-Pierre Se.
Table of contents
Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups. Prerequisites Topology and basic notions of measure theory.
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A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester. Some important basic results about the cohomology of Lie algebras include Whitehead's lemmas , Weyl's theorem , and the Levi decomposition theorem. The elements of the Chevalley—Eilenberg complex. The zeroth cohomology group is by definition the invariants of the Lie algebra acting on the module:.
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The first cohomology group is the space Der of derivations modulo the space Ider of inner derivations. From Wikipedia, the free encyclopedia. Archived from the original on Retrieved 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 . 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.
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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  ,  is by using the cochain complex , where is the module of all skew-symmetric -linear mappings , equipped with the coboundary acting by.
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.
Continuous groups in physics
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:.
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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  , . 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.
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Let be an ideal of and let be a -module. If and are free -modules, there exists a spectral sequence , with , converging to see  , . Similar spectral sequences exist for the homology . Further, let be a finite-dimensional Lie algebra over a field of characteristic 0, let be subalgebras such that is reductive in cf.