Difference between revisions of "Cohomology of Lie algebras"
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) {{MR|0148716}} {{MR|0143793}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} </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) {{MR|0874337}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) {{MR|0077480}} {{ZBL|0075.24305}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972) {{MR|0507234}} {{MR|0507236}} {{ZBL|0254.22005}} </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) {{MR|}} {{ZBL|0068.02102}} </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 {{MR|0024908}} {{ZBL|0031.24803}} </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) {{MR|0400275}} {{MR|0336651}} {{MR|0336650}} {{ZBL|}} </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) {{MR|}} {{ZBL|0256.22015}} </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 {{MR|0147577}} {{ZBL|0111.03302}} </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 {{MR|0142698}} {{MR|0142696}} {{ZBL|0134.03501}} </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 {{MR|0036511}} {{ZBL|0039.02901}} </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 {{MR|368073}} {{ZBL|}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A. Verona, "Introducere in coomologia algebrelor Lie" , Bucharest (1974) {{MR|0352221}} {{ZBL|0298.18006}} </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) {{MR|0644979}} {{ZBL|0464.22001}} </TD></TR></table> |
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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 {{MR|276285}} {{ZBL|0222.18019}} </TD></TR></table> |
Revision as of 17:32, 31 March 2012
A special case of cohomology of algebras. 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
[3]. 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. Deformation).
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 [6], [14]) is by using the cochain complex , where
is the module of all skew-symmetric
-linear mappings
, equipped with the coboundary
acting by
![]() |
![]() |
![]() |
![]() |
where the symbol means that the relevant argument is deleted. 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. 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:
![]() |
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 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. Let be an ideal of
and let
be a
-module. If
and
are free
-modules, there exists a spectral sequence
, with
, converging to
(see [3], [14]). Similar spectral sequences exist for the homology [3]. Further, let
be a finite-dimensional Lie algebra over a field
of characteristic 0, let
be subalgebras such that
is reductive in
(cf. Lie algebra, reductive), and let
be a semi-simple
-module. Then there exists a spectral sequence
, with
, converging to
(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 is a finite-dimensional semi-simple Lie algebra over such a field, the following results hold for every finite-dimensional
-module
:
![]() |
(Whitehead's lemma). The first of these properties is a sufficient condition for the semi-simplicity of a finite-dimensional algebra , and is equivalent to the semi-simplicity of all finite-dimensional
-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
is a reductive Lie algebra,
is a subalgebra of it and
is a finite-dimensional semi-simple module, then
, which reduces the calculation of the cohomology to the case of the trivial
-module
(see [5], [14]). The cohomology algebra
of a reductive Lie algebra
is naturally isomorphic to the algebra
of cochains invariant under
. In this case
is a Hopf algebra, and thus is an exterior algebra over the space
of primitive elements, graded in odd degrees
,
. In particular,
is the dimension of the centre of
, and
is isomorphic to the space of invariant quadratic forms on
(see [12], [14]). If
is algebraically closed, then
is the rank of the algebra
, that is, the dimension of its Cartan subalgebra
, and
are the degrees of the free generators in the algebra of polynomials over
invariant under
(or in the algebra of polynomials over
invariant under the Weyl group, which is isomorphic to it). In this case the numbers
are the dimensions of the primitive cohomology classes of the corresponding compact Lie group. The numbers
are called the exponents of the Lie algebra
. The homology algebra
of a reductive Lie algebra
over a field of characteristic 0 is the exterior algebra dual to
. For any
-dimensional Lie algebra
, an analogue of Poincaré duality holds:
![]() |
where and
is an arbitrary
-dimensional reductive subalgebra of
(see [14], ).
Only a few general assertions are known about the cohomology of solvable Lie algebras. For example, let be a finite-dimensional nilpotent Lie algebra over an infinite field and let
be a finite-dimensional
-module. Then
for all
if
has no trivial
-submodules, and
for
, and
for
if such a
-submodule does exist (see [7]). The groups
, are well-studied in the case that
is the nilpotent radical of the parabolic subalgebra
of some semi-simple Lie algebra
over an algebraically closed field of characteristic 0, and the representation of
in
is the restriction of some representation of
in
(see [11]). These cohomology groups are closely related to those of the complex homogeneous space
corresponding to the pair
, with values in sheaves of germs of holomorphic sections of homogeneous vector bundles over
. 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
![]() |
where is an ideal in
such that
is semi-simple [14].
In some cases, a relation can be established between the cohomology of Lie algebras and the cohomology of groups. Let be a connected real Lie group, let
be a maximal compact subgroup of it, let
be their Lie algebras, and let
be a finite-dimensional smooth
-module. If a natural
-module structure is defined on
, then
is isomorphic to the cohomology of
(as an abstract group), calculated by means of continuous cochains [10]. On the other hand, let
be the Lie algebra of a simply-connected solvable Lie group
, let
be a lattice in
and let
be a smooth finite-dimensional linear representation. If
is Zariski dense in the algebraic closure of
, then
(see [4]). In general,
. For nilpotent
it suffices to require that
be unipotent. If the lattice
in a simply-connected Lie group
is such that
is dense in the algebraic closure of the group
(for example, if
is nilpotent), then
.
In recent years there has been a systematic study of the cohomology of certain infinite-dimensional Lie algebras. Among these are the algebra of vector fields on a differentiable manifold
, 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.
References
[1] | N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201 |
[2] | D.B. [D.B. Fuks] Fuchs, "Cohomology of infinite-dimensional Lie algebras" , Plenum (1986) (Translated from Russian) MR0874337 |
[3] | H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) MR0077480 Zbl 0075.24305 |
[4] | M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972) MR0507234 MR0507236 Zbl 0254.22005 |
[5] | , Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Ecole Norm. Sup. (1955) Zbl 0068.02102 |
[6] | C. Chevalley, S. Eilenberg, "Cohomology theory of Lie groups and Lie algebras" Trans. Amer. Math. Soc. , 63 (1948) pp. 85–124 MR0024908 Zbl 0031.24803 |
[7] | J. Dixmier, "Cohomologie des algèbres de Lie nilpotents" Acta Sci. Mat. Szeged , 16 : 3–4 (1955) pp. 246–250 |
[8] | W. Greub, S. Halperin, R. Vanstone, "Connections, curvature and cohomology. Cohomology of principal bundles and homogeneous spaces" , 3 , Acad. Press (1975) MR0400275 MR0336651 MR0336650 |
[9] | P. de la Harpe, "Classical Banach–Lie algebras and Banach–Lie groups of operators in Hilbert space" , Springer (1972) Zbl 0256.22015 |
[10] | G. Hochschild, G.D. Mostow, "Cohomology of Lie groups" Ill. J. Math. , 6 : 3 (1962) pp. 367–401 MR0147577 Zbl 0111.03302 |
[11] | B. Kostant, "Lie algebra cohomology and the generalized Borel–Weil theorem" Ann. Math. , 74 : 2 (1961) pp. 329–387 MR0142698 MR0142696 Zbl 0134.03501 |
[12] | J.L. Koszul, "Homologie et cohomologie des algèbres de Lie" Bull. Soc. Math. France , 78 (1950) pp. 65–127 MR0036511 Zbl 0039.02901 |
[13] | 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 MR368073 |
[14] | A. Verona, "Introducere in coomologia algebrelor Lie" , Bucharest (1974) MR0352221 Zbl 0298.18006 |
[15] | A. Guichardet, "Cohomologie des groupes topologiques et des algèbres de Lie" , F. Nathan (1980) MR0644979 Zbl 0464.22001 |
Comments
The subcomplex of relative cochains is defined by
. Equivalently,
.
There is a generalization of the Poincaré duality result as follows. Let be free of finite dimension over
. For a
-module
let
be the dual Lie module defined by
for
,
,
, and let
be the
-module with underlying
-module
but with the
-action changed to
, where
is the action of
on
. Then there is a canonical isomorphism, [a1],
![]() |
of -modules where
. Note that if
is semi-simple then
.
References
[a1] | 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 MR276285 Zbl 0222.18019 |
Cohomology of Lie algebras. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomology_of_Lie_algebras&oldid=24057