Chevalley–Eilenberg Complex
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, Lie algebra cohomology is a
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s and
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s by relating cohomological methods of
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to properties of the Lie algebra. It was later extended by to coefficients in an arbitrary Lie module.


Motivation

If G is a
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Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra. This can be done as follows. Its cohomology is the
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of the complex of
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s on G. Using an averaging process, this complex can be replaced by the complex of left-invariant differential forms. The left-invariant forms, meanwhile, are determined by their values at the identity, so that the space of left-invariant differential forms can be identified with the
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of the Lie algebra, with a suitable differential. The construction of this differential on an exterior algebra makes sense for any Lie algebra, so it is used to define Lie algebra cohomology for all Lie algebras. More generally one uses a similar construction to define Lie algebra cohomology with coefficients in a module. If G is a simply connected ''noncompact'' Lie group, the Lie algebra cohomology of the associated Lie algebra \mathfrak g does not necessarily reproduce the de Rham cohomology of G. The reason for this is that the passage from the complex of all differential forms to the complex of left-invariant differential forms uses an averaging process that only makes sense for compact groups.


Definition

Let \mathfrak g be a Lie algebra over a commutative ring ''R'' with
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U\mathfrak g, and let ''M'' be a
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of \mathfrak g (equivalently, a U\mathfrak g-module). Considering ''R'' as a trivial representation of \mathfrak g, one defines the cohomology groups :\mathrm^n(\mathfrak; M) := \mathrm^n_(R, M) (see
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for the definition of Ext). Equivalently, these are the right
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s of the left exact invariant submodule functor :M \mapsto M^ := \. Analogously, one can define Lie algebra homology as :\mathrm_n(\mathfrak; M) := \mathrm_n^(R, M) (see
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for the definition of Tor), which is equivalent to the left derived functors of the right exact
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s functor : M \mapsto M_ := M / \mathfrak M. Some important basic results about the cohomology of Lie algebras include
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s,
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, and the
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theorem.


Chevalley–Eilenberg complex

Let \mathfrak be a Lie algebra over a
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k, with a left action on the \mathfrak-module M. The elements of the ''Chevalley–Eilenberg complex'' : \mathrm_k(\Lambda^\bullet\mathfrak,M) are called cochains from \mathfrak to M. A homogeneous n-cochain from \mathfrak to M is thus an alternating k-multilinear function f\colon\Lambda^n\mathfrak\to M. When \mathfrak is finitely generated as
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, the Chevalley–Eilenberg complex is canonically isomorphic to the
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M \otimes \Lambda^\mathfrak^*, where \mathfrak^*denotes the
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of \mathfrak. The Lie bracket cdot,\cdotcolon \Lambda^2 \mathfrak \rightarrow \mathfrak on \mathfrak induces a
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application d^_ \colon \mathfrak^* \rightarrow \Lambda^2 \mathfrak^* by duality. The latter is sufficient to define a derivation d_ of the complex of cochains from \mathfrak to k by extending d_^according to the graded Leibniz rule. It follows from the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
that d_ satisfies d_^2 = 0 and is in fact a differential. In this setting, k is viewed as a trivial \mathfrak-module while k \sim \Lambda^0\mathfrak^* \subseteq \mathrm(d_) may be thought of as constants. In general, let \gamma \in \mathrm(\mathfrak, \operatorname M) denote the left action of \mathfrak on M and regard it as an application d_\gamma^ \colon M \rightarrow M \otimes \mathfrak^*. The Chevalley–Eilenberg differential d is then the unique derivation extending d_\gamma^ and d_^ according to the
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, the nilpotency condition d^2 = 0 following from the Lie algebra homomorphism from \mathfrak to \operatorname M and the Jacobi identity in \mathfrak. Explicitly, the differential of the n-cochain f is the (n+1)-cochain df given by: :\begin (d f)\left(x_1, \ldots, x_\right) = &\sum_i (-1)^x_i\, f\left(x_1, \ldots, \hat x_i, \ldots, x_\right) + \\ &\sum_ (-1)^ f\left(\left _i, x_j\right x_1, \ldots, \hat x_i, \ldots, \hat x_j, \ldots, x_\right)\, , \end where the caret signifies omitting that argument. When G is a
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Lie group with Lie algebra \mathfrak, the Chevalley–Eilenberg complex may also be canonically identified with the space of left-invariant forms with values in M, denoted by \Omega^(G,M)^G. The Chevalley–Eilenberg differential may then be thought of as a restriction of the covariant derivative on the trivial
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G \times M \rightarrow G, equipped with the equivariant
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\tilde \in \Omega^1(G, \operatorname M) associated with the left action \gamma \in \mathrm(\mathfrak, \operatorname M) of \mathfrak on M. In the particular case where M = k = \mathbb is equipped with the trivial action of \mathfrak, the Chevalley–Eilenberg differential coincides with the restriction of the de Rham differential on \Omega^(G) to the subspace of left-invariant differential forms.


Cohomology in small dimensions

The zeroth cohomology group is (by definition) the invariants of the Lie algebra acting on the module: :H^0(\mathfrak; M) = M^ = \. The first cohomology group is the space of derivations modulo the space of inner derivations :H^1(\mathfrak; M) = \mathrm(\mathfrak, M)/\mathrm (\mathfrak, M)\, , where a derivation is a map d from the Lie algebra to M such that :d ,y= xdy-ydx~ and is called inner if it is given by :dx = xa~ for some a in M. The second cohomology group :H^2(\mathfrak; M) is the space of
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es of
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s :0\rightarrow M\rightarrow \mathfrak\rightarrow\mathfrak\rightarrow 0 of the Lie algebra by the module M. Similarly, any element of the cohomology group H^(\mathfrak; M) gives an equivalence class of ways to extend the Lie algebra \mathfrak to a "Lie n-algebra" with \mathfrak in grade zero and M in grade n. A Lie n-algebra is a
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with nonzero terms only in degrees 0 through n.


Examples


Cohomology on the trivial module

When M = \mathbb, as mentioned earlier the Chevalley–Eilenberg complex coincides with the de-Rham complex for a corresponding ''compact'' Lie group. In this case M carries the trivial action of \mathfrak, so xa = 0 for every x \in \mathfrak, a \in M. * The zeroth cohomology group is M. * First cohomology: given a derivation D, xDy = 0 for all x and y, so derivations satisfy D( ,y = 0 for all commutators, so the ideal mathfrak, \mathfrak/math> is contained in the kernel of D. **If mathfrak, \mathfrak= \mathfrak, as is the case for
simple Lie algebra In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of ...
s, then D \equiv 0, so the space of derivations is trivial, so the first cohomology is trivial. **If \mathfrak is
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group ...
, that is, mathfrak, \mathfrak= 0, then any linear functional D: \mathfrak \rightarrow M is in fact a derivation, and the set of inner derivations is trivial as they satisfy Dx = xa = 0 for any a \in M. Then the first cohomology group in this case is M^. In light of the de-Rham correspondence, this shows the importance of the compact assumption, as this is the first cohomology group of the n-torus viewed as an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
, and \mathbb^n can also be viewed as an abelian group of dimension n, but \mathbb^n has trivial cohomology. * Second cohomology: The second cohomology group is the space of equivalence classes of central extensions :0 \rightarrow \mathfrak \rightarrow \mathfrak \rightarrow \mathfrak \rightarrow 0. Finite dimensional, simple Lie algebras only have trivial central extensions: a proof is provided
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.


Cohomology on the adjoint module

When M = \mathfrak, the action is the
adjoint action In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is \m ...
, x\cdot y = ,y= \text(x)y. * The zeroth cohomology group is the center \mathfrak(\mathfrak) * First cohomology: the inner derivations are given by Dx = xy = ,y= -\text(y)x, so they are precisely the image of \text: \mathfrak \rightarrow \operatorname \mathfrak. The first cohomology group is the space of
outer derivation {{Short pages monitor