A connected
Lie group
that does not contain non-trivial connected solvable
(or, equivalently, connected Abelian) normal subgroups. A connected Lie group
is semi-simple if and only if its Lie algebra is semi-simple (cf.
Lie algebra, semi-simple).
A connected Lie group
is said to be
simple
if its Lie algebra is simple, that is, if
does not contain non-trivial connected normal subgroups other than
.
A connected Lie group is semi-simple if and only if it
splits into a locally direct product of simple non-Abelian normal subgroups.
The classification of semi-simple Lie groups reduces to the local
classification, that is, to the classification of semi-simple Lie algebras (cf.
Lie algebra, semi-simple),
and also to the global classification of the Lie groups
that correspond to a given semi-simple Lie algebra
.
In the case of Lie groups over the field
of complex numbers the main result of the local classification is
that every simply-connected simple non-Abelian complex Lie group
is isomorphic to one of the groups
,
,
,
(the universal covering of the group
),
,
(see
Classical group),
or one of the exceptional complex Lie groups (see
Lie algebra, exceptional).
The global classification of the Lie groups corresponding to a semi-simple Lie algebra
over
goes as follows. Let
be a
Cartan subalgebra
of
and let
be the
root system
of
with respect to
.
To every semi-simple Lie group
with Lie algebra
corresponds a lattice
that is the kernel of the
exponential mapping
.
In particular, if
is simply connected, then
coincides with the lattice
generated by the elements
,
(see
Lie algebra, semi-simple),
and if
is a group without centre (an
adjoint group),
then
is the lattice
In the general case
.
For any additive subgroup
satisfying the condition
there is a unique (up to isomorphism) connected Lie group
with Lie algebra
such that
.
The centre of
is isomorphic to
,
and for the
fundamental group
one has:
The quotient group
(the centre of the simply-connected Lie group with Lie algebra
)
is finite and for the different types of simple Lie algebras
it has the following form:
The order of the group
is the same as the number of vertices with coefficient 1 in the extended
Dynkin diagram
of
;
discarding one of the vertices gives the Dynkin diagram.
A similar classification holds for compact real semi-simple Lie groups,
each of which is imbedded in a unique complex
semi-simple Lie group as a maximal compact subgroup (see
Lie group, compact).
The global classification of non-compact real semi-simple Lie groups can be
carried out in a similar but more complicated way. In particular, the centre
of the simply-connected Lie group corresponding to a semi-simple Lie algebra
over
can be calculated as follows. Let
be the
Cartan decomposition,
where
is a maximal compact subalgebra of
and
is its orthogonal complement with respect to the Killing form, let
be the corresponding involutive automorphism, extended to
,
the Cartan subalgebra of
containing a Cartan subalgebra
,
an automorphism of
that coincides with
on the roots with respect to
and extended to the root vectors in an appropriate way, and
the Cartan decomposition of the real form
corresponding to
.
Then
(see
[3],
where this group is calculated for all types of simple algebras
over
).
Every complex semi-simple Lie group
has the unique structure of an affine
algebraic group compatible with the analytic structure specified
on it, and any analytic homomorphism of
to an algebraic group is rational. The corresponding algebra of regular functions on
coincides with the algebra of holomorphic representation functions.
On the other hand, a non-compact real semi-simple Lie
group does not always admit a faithful linear representation
— the simplest example is the simply-connected
Lie group corresponding to the Lie algebra
.
If
is a semi-simple Lie algebra over
,
then in the centre
of the simply-connected group
corresponding to
there is a smallest subgroup
,
called the
linearizer,
such that
is isomorphic to a linear semi-simple Lie group. If
is the compact real form of
,
then
(see
[3],
where this group is calculated for all types of simple Lie algebras
).