Automorphisms of simple Chevalley groups over $\bold Q$-algebras

Automorphisms and Isomorphisms of Chevalley Groups and Algebras

Suppose that Φ is a reduced irreducible root system, R is an associative commutative ring with unity, G(Φ, R) is the corresponding adjoint Chevalley group, and E(Φ, R) is its elementary subgroup (see Section 5). There are a lot of results (see, e.g., [Wat80], [Pet82], [GMi83], [HO’M89], [Abe93], [Che00], [Bun07], and references therein*) asserting that, under some conditions, all automorphisms ...

Chevalley Groups over Commutative Rings

Torsion automorphisms of simple Lie algebras

An automorphism σ of a simple finite dimensional complex Lie algebra g is called torsion, if σ has finite order in the group Aut(g) of all automorphisms of g. The torsion automorphisms of g were classified by Victor Kac in [12], as an application of his results on infinite dimensional Lie algebras. Those torsion automorphisms contained in the identity component G = Aut(g)◦ are called inner; the...

Boundedly Simple Groups of Automorphisms of Trees

A group is boundedly simple if for some constant N , every nontrivial conjugacy class generates the whole group in N steps (bounded simplicity implies simplicity). For a large class of trees, Tits proved simplicity of automorphism groups generated by stabilisers of edges. We determine the subclass of trees for which such groups are boundedly simple. This is the class of subdivisions of bi-regul...

Infinite Dimensional Chevalley Groups and Kac-moody Groups over Z

Let A be a symmetrizable generalized Cartan matrix. Let g be the corresponding Kac-Moody algebra. Let G(R) be Tits’ Kac-Moody group functor over commutative rings R associated to g. Then G(R) is given by five axioms KMG1-KMG5 which are natural extensions of the properties of Chevalley-Demazure group schemes. We give a construction of the Tits functor for symmetrizable Kac-Moody groups G using i...