Semi-simple algebraic group
A connected linear algebraic group of positive dimension which contains only trivial solvable (or, equivalently, Abelian) connected closed normal subgroups. The quotient group of a connected non-solvable linear group by its radical is semi-simple.
A connected linear algebraic group of positive dimension is called simple (or quasi-simple) if it does not contain proper connected closed normal subgroups. The centre
of a simple group
is finite, and
is simple as an abstract group. An algebraic group
is semi-simple if and only if
is a product of simple connected closed normal subgroups.
If the ground field is the field of complex numbers, a semi-simple algebraic group is nothing but a semi-simple Lie group over
(cf. Lie group, semi-simple). It turns out that the classification of semi-simple algebraic groups over an arbitrary algebraically closed field
is analogous to the case
, that is, a semi-simple algebraic group is determined up to isomorphism by its root system and a certain sublattice in the weight lattice that contains all the roots. More precisely, let
be a maximal torus in the semi-simple algebraic group
and let
be the character group of
, regarded as a lattice in the space
. For a rational linear representation
of
, the group
is diagonalizable. Its eigenvalues, which are elements of
, are called the weights of the representation
. The non-zero weights of the adjoint representation
are called the roots of
. It turns out that the system
of all roots of
is a root system in the space
, and that the irreducible components of the system
are the root systems for the simple closed normal subgroups of
. Furthermore,
, where
is the lattice spanned by all roots and
is the weight lattice in the root system
. In the case
the space
can be naturally identified with a real subspace
, where
is the Lie algebra of the torus
, spanned by the differentials of all characters, while the lattices in
dual to
coincide (up to a factor
) with
(see Lie group, semi-simple).
The main classification theorem states that if is another semi-simple algebraic group,
its maximal torus,
a root system of
, and if there is a linear mapping
giving an isomorphism between the root systems
and
and mapping
onto
, then
(local isomorphism). Moreover, for any reduced root system
and any lattice
satisfying the condition
there exists a semi-simple algebraic group
such that
is its root system with respect to the maximal torus
, and
.
The isogenies (in particular, all automorphisms, cf. Isogeny) of a semi-simple algebraic group have also been classified.
References
[1] | R.G. Steinberg, "Lectures on Chevalley groups" , Yale Univ. Press (1968) |
[2] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) |
Comments
References
[a1] | T.A. Springer, "Linear algebraic groups" , Birkhäuser (1981) |
Semi-simple algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-simple_algebraic_group&oldid=18012