On the adjoint quotient of Chevalley groups over arbitrary base schemes

On the adjoint quotient of Chevalley groups over arbitrary base schemes

For a split semisimple Chevalley group scheme G with Lie algebra g over an arbitrary base scheme S, we consider the quotient of g by the adjoint action of G. We study in detail the structure of g over S. Given a maximal torus T with Lie algebra t and associated Weyl group W , we show that the Chevalley morphism π : t/W → g/G is an isomorphism except for the group Sp2n over a base with 2-torsion...

Chevalley Groups over Commutative Rings

Lectures on Chevalley Groups

( b ) H is maximal n i l p o t e n t and every subgroup of f i n i t e l a d e x . . i s of f i n i t e index i n i t s normalizer. Proof of Theorem 6: ( a ) Map Ga -> #, by x,: t -> x,(t) . This is a r a t i o n a l homomorphism. So s i n c e Ga i s a connected , ~ g e b r a i c group s o i s Xa . Hence G is a lgebra ic and connected.. Let R = r a d G . Since R is so lvable and normal it i s f...

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...

Algebraic Groups over Arbitrary Fields

In this chapter, we study algebraic groups, especially nonsplit reductive groups, over arbitrary fields. The algebraic groups over a field k that become isomorphic to a fixed algebraic group over kal are classified by a certain cohomology group. In the first section, we explain this, and discuss what is known about the cohomology groups. Over an algebraically closed field, the classical semisim...