On Some Simple Groups Defined by C. Chevalley

Abstract

RIMHAK REEIntroduction.For any semi-simple Lie algebra g over the field of complex numbers and an arbitrary field K, C. Chevalley [2] defined a group by a uniform method.The group turns out to be simple when g is simple (except for a few exceptional cases), and yields some new classes of simple groups not contained in the theory of classical groups, if, in particular, g is one of the exceptional simple Lie algebras.In this paper first we consider the case where g is one of the classical algebras An, Bn, C", Dn, and identify the group with classical groups, thus answering a question raised in the last section of [2].Our results are as follows:(a) if g is of the type An, then is the special projective group PSL(+1, K);(b) if g is of the type Cn, then is the quotient group of the symplectic group Sp(2ra, K) over its center;(c) if g is of the type >", then is the commutator group of the projective orthogonal group defined by the quadratic form 5^**=J *-* (3 gra);(d) if g is of the type Bn and if K is not of characteristic 2 then is the commutator group of the projective orthogonal group defined by the quadratic form yiLn *_* (2 gra); if g is of type Bn and if K is of characteristic 2 then is a subgroup of Sp (2ra, K); if, moreover, K is perfect(x) then G = Sp(2ra, K).The author has been unable to identify in case K is not perfect and of characteristic 2. Secondly, we consider the case where g is the exceptional algebra G2, and