On cubic action of a rank one group

Details On cubic action of a rank one group On cubic action of a rank one group

2011-06-12

arxiv.org/abs/1106.2310v2

Mathematics

Group Theory

39 pages, Corollary 7.6 added

Scientific paper

We consider a rank one group $G=< A, B > $ which acts cubically on a module $V$, this means $[V,A,A,A] =0$ but $[V,G,G,G] \ne 0$. We assume that $A_0:=C_A([V,A]) \cap C_A(V/C_V(A)) $ is not trivial; this is always the case if $A$ is not abelian. Then $A_0$ defines a subgroup $G_0$ of $G$ which acts quadratically on $V$. By a theorem of Timmesfeld $G_0 \cong SL_2(J,R)$ for a ring $R$ and a special quadratic Jordan 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 $\pi$ with Witt index 1.

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