In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Lie algebra cohomology is a
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
theory for
Lie algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. It was first introduced in 1929 by
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
to study the topology of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s and
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
s by relating cohomological methods of
Georges de Rham
Georges de Rham (; 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology.
Biography
Georges de Rham was born on 10 September 1903 in Roche, a small village in the canton of Vaud in ...
to properties of the Lie algebra. It was later extended by to coefficients in an arbitrary
Lie module.
Motivation
If
is a compact
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
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
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
of the complex of
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s on
. 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
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
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
is a simply connected ''noncompact'' Lie group, the Lie algebra cohomology of the associated Lie algebra
does not necessarily reproduce the de Rham cohomology of
. 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
be a
Lie algebra over a commutative ring ''R'' with
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representati ...
, and let ''M'' be a
representation of
(equivalently, a
-module). Considering ''R'' as a trivial representation of
, one defines the cohomology groups
:
(see
Ext functor
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic stru ...
for the definition of Ext). Equivalently, these are the right
derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in vari ...
s of the left exact invariant submodule functor
:
Analogously, one can define Lie algebra homology as
:
(see
Tor functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to constr ...
for the definition of Tor), which is equivalent to the left derived functors of the right exact
coinvariant
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
s functor
:
Some important basic results about the cohomology of Lie algebras include
Whitehead's lemmas,
Weyl's theorem In mathematics, Weyl's theorem or Weyl's lemma might refer to one of a number of results of Hermann Weyl. These include
* the Peter–Weyl theorem
* Weyl's theorem on complete reducibility, results originally derived from the unitarian trick on r ...
, and the
Levi decomposition
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semi ...
theorem.
Chevalley–Eilenberg complex
Let
be a Lie algebra over a field
, with a left action on the
-module
. The elements of the ''Chevalley–Eilenberg complex''
:
are called cochains from
to
. A homogeneous
-cochain from
to
is thus an alternating
-multilinear function
. When
is finitely generated as vector space, the Chevalley–Eilenberg complex is canonically isomorphic to the tensor product
, where
denotes the dual vector space of
.
The Lie bracket
on
induces a
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
application
by duality. The latter is sufficient to define a derivation
of the complex of cochains from
to
by extending
according to the graded Leibniz rule. It follows from the Jacobi identity that
satisfies
and is in fact a differential. In this setting,
is viewed as a trivial
-module while
may be thought of as constants.
In general, let
denote the left action of
on
and regard it as an application
. The Chevalley–Eilenberg differential
is then the unique derivation extending
and
according to the
graded Leibniz rule
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
__TOC__
Definition
A differential graded a ...
, the nilpotency condition
following from the Lie algebra homomorphism from
to
and 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 asso ...
in
.
Explicitly, the differential of the
-cochain
is the
-cochain
given by:
where the caret signifies omitting that argument.
When
is a real Lie group with Lie algebra
, the Chevalley–Eilenberg complex may also be canonically identified with the space of left-invariant forms with values in
, denoted by
. The Chevalley–Eilenberg differential may then be thought of as a restriction of the covariant derivative on the trivial
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
, equipped with the equivariant
connection associated with the left action
of
on
. In the particular case where
is equipped with the trivial action of
, the Chevalley–Eilenberg differential coincides with the restriction of the
de Rham differential on
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:
:
The first cohomology group is the space of derivations modulo the space of inner derivations
:
,
where a derivation is a map
from the Lie algebra to
such that
:
and is called inner if it is given by
:
for some
in
.
The second cohomology group
:
is the space of equivalence classes of
Lie algebra extensions
:
of the Lie algebra by the module
.
Similarly, any element of the cohomology group
gives an equivalence class of ways to extend the Lie algebra
to a "Lie
-algebra" with
in grade zero and
in grade
.
A Lie
-algebra is a
homotopy Lie algebra In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L_\infty-algebra) is a generalisation of the concept of a differential graded Lie algebra. To be a little more specific, the Jacobi identity only holds up to ho ...
with nonzero terms only in degrees 0 through
.
Examples
Cohomology on the trivial module
When
, as mentioned earlier the Chevalley–Eilenberg complex coincides with the de-Rham complex for a corresponding ''compact'' Lie group. In this case
carries the trivial action of
, so
for every
.
* The zeroth cohomology group is
.
* First cohomology: given a derivation
,
for all
and
, so derivations satisfy
for all commutators, so the ideal