On cubic action of a rank one group

Abstract

We consider a rank one group $G = \langle A,B \rangle $ which acts cubically on a module $V$, this means $[V,A,A,A] =0$ but $[V,G,G,G] \ne 0$. We have to distinguish whether the group $A_0 :=C_A([V,A]) \cap C_A(V/C_V(A))$ is trivial or not. We show that if $A_0$ is trivial, $G$ is a rank one group associated to a quadratic Jordan division algebra. If $A_0$ is not trivial (which is always the case if $A$ is not abelian), then $A_0$ defines a subgroup $G_0$ of $G$ which acts quadratically on $V$. We will call $G_0$ the \textit{quadratic kernel} of $G$. By a result of Timmesfeld we have $G_0 \cong \SL_2(J,R)$ for a ring $R$ and a special quadratic Jordan division algebra $J \subseteq R$. 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 $\pi$ 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 $\characteristic V\ne 2,3$, then $G$ is a unitary group or an exceptional algebraic group.