Chevalley group

A linear algebraic group over some field, related to a semi-simple complex Lie algebra. Let be a semi-simple algebra over , let be its Cartan subalgebra, let be a root system of with respect to , let be a system of simple roots, let be a Chevalley basis of the algebra , and let be its linear envelope over . Let be a faithful representation of the Lie algebra in a finite-dimensional vector space . It turns out that there is a lattice in (i.e. a free Abelian subgroup a basis of which is the basis of the space ) which is invariant with respect to all operators (, is a natural number). If is an arbitrary field and if , then homomorphisms , , are defined and are given by the formulas

The subgroups , , generate in some subgroup , which is called the Chevalley group related to the Lie algebra , the representation and the field . If (the adjoint representation), the Chevalley groups were defined by C. Chevalley in 1955 (see [1]).

If is an algebraically closed field containing , then a Chevalley group is a connected semi-simple linear algebraic group over , defined and split over the prime subfield . Its Lie algebra is isomorphic to . The group is the commutator subgroup of the group of points of that are rational over . Any connected semi-simple linear algebraic group over is isomorphic to one of the Chevalley groups. The algebraic groups (and as abstract groups) depend only on the lattice generated by the weights of the representation . If coincides with the lattice of roots , then is called the adjoint group, and if (the lattice of weights, see Lie group, semi-simple), then is called a universal or simply-connected group. If is universal, then .

The Chevalley group always coincides with its commutator subgroup. The centre of is finite. For example, the centre of the universal group is isomorphic to , and the corresponding adjoint group is isomorphic to and has trivial centre.

If the algebra is simple, then the adjoint Chevalley group is simple, except in the following cases: and is a Lie algebra of type or ; or and is a Lie algebra of type . Other series of simple groups can be obtained when considering the subgroups of fixed points of certain automorphisms of finite order of Chevalley groups (so-called torsion groups).

If the field is finite, then the order of the universal group can be computed by the formula

where , () are exponents of the Lie algebra , i.e. the degrees of the free polynomials on , generating the algebras, that are invariant with respect to the Weyl group, and is the number of positive roots.

There is a well-developed theory of rational linear representations of the Chevalley groups over an infinite field . It is reduced to the case of an algebraically closed field, and in the latter case coincides with the theory of rational representations (cf. Rational representation) of semi-simple algebraic groups. If is simple, is the universal Chevalley group over the infinite field , and is a non-trivial irreducible finite-dimensional representation of (as an abstract group) over an algebraically closed field , then there exists a finite set of imbeddings and a set of rational representations of the groups such that . Concerning representations of Chevalley groups, see also [2], [3], [5].

References

[1]	C. Chevalley, "Sur certains groupes simples" Tôhoku Math. J. , 7 : 1–2 (1955) pp. 14–66 MR0073602 Zbl 0066.01503
[2]	R.G. Steinberg, "Lectures on Chevalley groups" , Yale Univ. Press (1968) MR0466335 Zbl 1196.22001
[3]	A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.) , Seminar on algebraic groups and related finite groups , Lect. notes in math. , 131 , Springer (1970)
[4]	J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) MR0323842 Zbl 0254.17004
[5]	J.E. Humphreys, "Ordinary and modular representations of Chevalley groups" , Springer (1976) MR0453884 Zbl 0341.20037

Comments

In the above denotes the characteristic of the field .

The torsion groups are also called the twisted Chevalley groups or the Steinberg groups. They were introduced by R. Steinberg in [a1].

An important reference for the representation theory of Chevalley groups is the recent textbook by R.W. Carter [a2].

References

[a1]	R. Steinberg, "Variations on a theme of Chevalley" Pacific J. Math. , 9 (1959) pp. 875–891 MR0109191 Zbl 0092.02505
[a2]	R.W. Carter, "Finite groups of Lie type: Conjugacy classes and complex characters" , Wiley (Interscience) (1985)