Cubic Action of a Rank One Group

Cubic Action of a Rank One Group Synopsis

We consider a rank one group G = A,Bacting 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 one group 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 group or an exceptional algebraic group.