E-raamat: Cubic Action of a Rank One Group

"We consider a rank one group G = A,B acting cubically on a module V , this means [ V, A, A,A] = 0 but [ V, G, G,G] = 0. We have to distinguish whether the group A0 := CA([ V,A]) CA(V/CV (A)) is trivial or not. We show that if A0 is trivial, G is a rank onegroup associated to a quadratic Jordan division algebra. If A0 is not trivial (which is always the case if A is not abelian), then A0 defines a subgroup G0 of G acting quadratically on V . We will call G0 the quadratic kernel of G. By a result of Timmesfeld we have G0 = SL2(J,R) for a ring R and a special quadratic Jordan division algebra J R. We show that J is either a Jordan algebra contained in a commutative field or a Hermitian Jordan algebra. In the second case G is the special unitary group of a pseudo-quadratic form of Witt index 1, in the first case G is the rank one group for a Freudenthal triple system. These results imply that if (V,G) is a quadratic pair such that no two distinct root groups commute and charV = 2, 3, then G is a unitary groupor an exceptional algebraic group"--