A group
having the structure of an
analytic manifold
such that the mapping
of the direct product
into
is analytic. In other words, a Lie group is a
set endowed with compatible structures of a group and an analytic
manifold. A Lie group is said to be real, complex or
-adic,
depending on the field over which its analytic manifold is
considered. Henceforth, as a rule, real Lie groups are considered (every complex
Lie group is naturally endowed with the structure of a
real Lie group by means of restriction of the ground field; for Lie groups over
-adic
number fields see
Lie group,
-adic;
Analytic group).
Examples of Lie groups.
The
general linear group
over the field
of real numbers (see also
Linear group)
and its subgroups, closed in the natural Euclidean topology
(J. von Neumann,
1927).
The main concepts of the theory of Lie groups were introduced into
mathematics in the
1870-s by
S. Lie.
Lie groups arose in
connection with the problem of the solvability of differential equations
by quadratures and research into continuous transformation groups. The successful
application of group theory to the solution of algebraic equations
of higher degrees, which manifested itself in the creation of
Galois theory,
suggested an attempt to construct an analogue of Galois
theory for differential equations. Although groups took up a
position in the theory of differential equations somewhat
different from that in the theory of algebraic equations, this led to the creation
of the theory of Lie groups, and also to the theory of
algebraic groups, which has deep connections with many branches of mathematics.
Lie groups were originally defined as local transformation groups of the
-dimensional
space
(or
)
that depend analytically on a finite system of parameters, and it
was required that the parameters of a product of transformations be expressible
in terms of the parameters of the factors by means
of analytic functions. Later on, mathematicians turned to the abstract
consideration of Lie groups, but also from the local point of view (see
Lie group, local).
Systematic research into the global structure of Lie groups was first begun by
E. Cartan
and
H. Weyl.
The first modern account of the theory of
Lie groups was given in
1938
by
L.S. Pontryagin
(see
[1]).
Does the replacement of analyticity of the manifold
and the mapping
by differentiability lead to an extension of the class
of Lie groups? This question was solved by Lie: If
is twice continuously differentiable, then
is a Lie group.
Hilbert's fifth problem
turned out to be considerably more complicated: If
is an
-dimensional
topological manifold and the mapping
is continuous, is
a Lie group? For compact groups this problem was affirmatively
solved by von Neumann in
1933,
and for locally compact
Abelian groups by Pontryagin in
1934.
In the general case
an affirmative answer was obtained in
1952
by
A.M. Gleason,
D. Montgomery
and
L. Zippin
(see
[4],
and also
[18]).
Thus one can define a Lie group as a
topological group
whose topological space is a finite-dimensional (or locally Euclidean)
manifold.
This is very important for the general theory of topological groups.
A subset
of a Lie group
is called a
subgroup
(more precisely, a
Lie subgroup)
if
is a subgroup of the abstract group
and a submanifold of the analytic manifold
.
A
morphism
between Lie groups
and
is an
analytic mapping
that is a
homomorphism
of abstract groups; if
is also bijective and
is analytic, then
is called an
isomorphism of Lie groups;
in the case when
is locally bijective (around the identity
)
one says that the Lie groups
and
are
locally isomorphic.
The
dimension of a Lie group
is the dimension of
as an analytic manifold. Henceforth only finite-dimensional Lie groups are
considered, although many results can be generalized to
the case of Banach Lie groups (cf.
Lie group, Banach).
Let
be a closed normal subgroup of a finite-dimensional Lie group
.
Then the quotient group
can be endowed with the structure of an analytic manifold such that
becomes a Lie group and the canonical mapping
is a morphism.
The correspondence between Lie groups and Lie algebras.
The main method of research in the theory of Lie groups is
the infinitesimal method created by Lie. This method makes it possible
to reduce the study of such a complicated object as a Lie
group largely to the study of a purely algebraic object, a
Lie algebra.
To every Lie group
corresponds a Lie algebra
,
constructed as follows (see also
Lie algebra of an analytic group).
A
left-invariant vector field
on
is a vector field that is invariant with respect
to the differentials of left translations, that is,
is a left-invariant vector field if
for any
,
where
.
The left-invariant vector fields on
form a vector space that can be identified with the tangent space
at the identity
of the group
,
by associating with the field
its value at
.
If
,
then the
Lie bracket
is also a left-invariant field, and this defines in
a bilinear operation with respect to which
becomes the Lie algebra
(here
denotes composition of vector fields, regarded as
derivations of the algebra of infinitely-differentiable
real-valued functions on the manifold
).
A more explicit construction of the commutation operation
in
is as follows: Let
be the integral curves of the fields
in
passing through the identity
of the group. Then
is the tangent vector at
to the curve
 |
Recovering a Lie group
from its Lie algebra
is possible by the
exponential mapping
,
which associates with a field
the element
of its integral curve
.
If
is a linear Lie group, that is, a subgroup of the
general linear group
,
then
can be identified with a subalgebra of the general matrix Lie algebra
and the exponential mapping takes the form
The mapping
is analytic and a local isomorphism, and so in
some neighbourhood of the identity of the group
it defines a local chart
(
canonical coordinates).
According to the
Campbell–Hausdorff formula
the multiplication in
in canonical coordinates, that is, the locally defined mapping
can be given in terms of operations in the Lie algebra
.
Thus, locally a Lie group is completely determined by its Lie algebra.
The correspondence between Lie groups and Lie algebras has deep functorial
properties. A Lie group is determined by its Lie algebra up
to a local isomorphism; in particular, if two Lie groups
and
are connected and simply connected, then the isomorphy of their Lie algebras implies
.
Arcwise-connected subgroups of a Lie group
correspond one-to-one to subalgebras of the Lie algebra
.
Let
be a morphism of Lie groups. Then the differential of this
morphism at the identity is a homomorphism of Lie algebras:
In general, not every homomorphism
has the form
,
but if
is simply connected this is the case. An arcwise-connected subgroup
of a connected Lie group
is normal if and only if
is an
ideal
of the Lie algebra
;
if in addition
is closed in
,
then
By construction, the Lie algebra
of a given Lie group
is an analytic invariant. In reality
is a topological invariant; this follows immediately from the following
theorem of Cartan:
A continuous homomorphic mapping of a (real) Lie group
into a Lie group
is a morphism. For complex Lie groups this assertion
is not always true, although it holds for
-adic
Lie groups (see
[3]).
The automorphism group
of a connected Lie group
is a Lie group which can be identified with a Lie subgroup of
.
In particular, if the Lie group
is simply-connected, then
where
denotes the Lie algebra of derivations of the algebra
.
The correspondence
where
is the inner automorphism implemented by the element
,
is called the
adjoint representation of the Lie group
;
its differential is the adjoint representation
of the Lie algebra
.
Cf. also
Adjoint representation of a Lie group.
The global structure of Lie groups.
The existence of a global Lie group with a given
real Lie algebra was proved in
1930
by Cartan. He also showed
that a closed subgroup of a real Lie group is a Lie subgroup. Two
types of Lie groups play a special role, namely: semi-simple and solvable ones (see
Lie group, semi-simple;
Lie group, solvable).
A connected Lie group
is said to be
semi-simple
if it does not contain connected solvable normal subgroups other than the identity; if
does not contain non-trivial connected normal subgroups, it is said to be
simple.
The Lie algebra
of a semi-simple, simple or solvable Lie group
is, respectively, a semi-simple, simple or solvable Lie algebra. The
investigation of arbitrary Lie groups essentially reduces to the study
of semi-simple and solvable Lie groups. Any Lie group
has a largest connected solvable normal subgroup, called the
solvable radical
and denoted by
.
In
there are maximal semi-simple subgroups. If
is one of them, then
,
and all maximal semi-simple subgroups are conjugate; if
is simply connected, then
and the product is semi-direct (the
Levi–Mal'tsev theorem).
The existence of this decomposition was first proved by
E. Levi
in
1905
for complex Lie algebras, and the conjugacy of the semi-simple
components was established by
A.I. Mal'tsev
in
1942
(see
[16],
[3],
and also
Levi–Mal'tsev decomposition).
The most general fact about solvable Lie groups was obtained
by Lie: Any connected solvable linear group over the field
can be transformed to triangular form; that is, the description of connected
solvable Lie groups reduces to the description
of subgroups of the general triangular group
.
A detailed investigation of solvable subgroups was undertaken by Mal'tsev in
[16].
In the study of the structure of semi-simple Lie groups an
important role is played by their maximal compact subgroups, studied by
Cartan in close connection with the theory of symmetric spaces (see
[10]).
According to Cartan's classical theorem, maximal compact
subgroups of a semi-simple Lie group
are conjugate; if
is a maximal compact subgroup of
,
there is a submanifold
,
analytically isomorphic to a Euclidean space, such that
and the mapping
,
,
is an isomorphism of analytic manifolds. Thus, the topological structure of
is determined by the topological structure of
.
Mal'tsev
[16]
extended Cartan's theorem to arbitrary connected Lie groups. Another decomposition
of a connected Lie group into a product of a maximal
compact subgroup and a Euclidean space was found by
K. Iwasawa
(see
Iwasawa decomposition).
Linear representability.
From the very beginning of the development of the theory of Lie groups
it was clear that arbitrary Lie groups are close to linear Lie
groups. Lie proved that in many cases Lie groups
are locally isomorphic to linear Lie groups. The general
theorem
was obtained by
I.D. Ado
in
1935:
Any Lie group
is locally isomorphic to a linear Lie group (see
[15]).
At the same time it is not difficult to give
examples of Lie groups that are not linear, e.g. the simply-connected covering group of
,
or (in the case of the field
)
a complex compact torus. If
is a simply-connected solvable Lie group, then any Lie subgroup of it
is simply connected and isomorphic to a linear Lie group. In
the general case the following criterion has been found for linear representability
[16]:
A connected Lie group
is linear if and only if its radical
and semi-simple quotient group
are linear; in turn, for
to be linearly representable it is necessary and
sufficient that its commutator subgroup should be simply connected,
and the linearity of the semi-simple Lie group
depends on the structure of its centre. Compact, and also
complex semi-simple, Lie groups are not only
linear, but also linear algebraic groups (cf.
Linear algebraic group)
.
Classification.
One of the main problems in the theory of Lie groups is
that of classifying arbitrary connected Lie groups up to isomorphism. In the
class of all locally isomorphic connected Lie groups that have the
same Lie algebra there is a unique simply-connected Lie group
,
and any Lie group
of this class is isomorphic to
where
is a discrete central normal subgroup. Therefore, the classification
of Lie groups reduces to the classification of finite-dimensional
Lie algebras and the calculation of the centres of simply-connected
Lie groups. On the other hand, it reduces to
the classification of two fundamentally different types
of groups: semi-simple and solvable (see
Lie group, semi-simple,
Lie group, solvable).
At first glance solvable Lie groups have a simpler structure and
their classification, it would seem, should not be difficult. However, this impression
is deceptive and up to now
(1988)
there is no hope
of obtaining a classification of solvable Lie groups. By contrast,
semi-simple Lie groups have been completely classified. A complete
classification of complex semi-simple Lie algebras was obtained
by
W. Killing
in
1888–1890
(see
[1],
[3]).
Since a complex semi-simple Lie algebra is a direct
sum of simple subalgebras, it is sufficient to classify simple
Lie algebras. It turns out that there are only nine types
of complex simple Lie algebras, namely the four infinite series
 |
and the five exceptional algebras
(see also
Lie algebra, semi-simple).
To the infinite series of complex simple Lie algebras correspond the
classical linear Lie groups. The corresponding
simply-connected groups have the form: type
—
;
type
—
,
the
spinor group
corresponding to a non-singular quadratic form
of dimension
;
type
— the
symplectic group
of degree
;
type
—
.
It is not difficult to compute the centres of these groups. For example, the centre of
is the cyclic group of order
,
and the centres of
and the symplectic group are the cyclic group of order 2. In
this way one obtains a classification of complex semi-simple Lie groups. The
classification of real semi-simple Lie groups turns out to be much
more complicated and depends on the classification of their real forms (cf. also
Form of an algebraic group).
The most important point here is the existence for any complex semi-simple group
of a unique compact real form
;
this implies that the Lie algebra
is isomorphic to
,
that is, it is obtained by complexifying the Lie algebra
.
Relying on this, Cartan in
1914
obtained a
complete classification of the real forms of complex semi-simple Lie groups. In terms of
Galois cohomology
this is equivalent to a description of the set
(see also
Linear algebraic group).
Later Killing's method was perfected by Cartan and Weyl, which
gave the possibility of solving a number of other classification
problems, and also to develop the important theory of representations
of Lie groups. A classification of semi-simple subgroups of the
classical complex simple Lie groups has been obtained (see
[17]).
Modern development and applications.
In the
1950-s a new step in the development of the theory
of Lie groups was begun, which manifested itself, in particular,
in the creation of the theory of algebraic groups (see
Linear algebraic group).
Earlier
C. Chevalley
(see
)
had explained in detail the algebraic nature of
the fundamental results of Lie group theory. The application
of methods of algebraic geometry made it possible to illuminate these
classical results in a new way and revealed new deep
connections with the theory of functions, number theory, etc. The theory of
-adic
Lie groups (cf.
Lie group,
-adic)
had a significant development (see
[3],
[6]).
Lie groups are connected in practice with all
main branches of mathematics: with geometry and topology
through the theory of Lie transformation groups (cf.
Lie transformation group),
with analysis through the theory of linear representations, etc. The various
applications of Lie groups to physics and mechanics are also extremely important.