Cubic Action of a Rank one Group

Abstract

We consider a rank one group G = ⟨ A , B ⟩ G = \langle A,B \rangle acting cubically on a module V V , this means [ V , A , A , A ] = 0 [V,A,A,A] =0 but [ V , G , G , G ] ≠ 0 [V,G,G,G] \ne 0 . We have to distinguish whether the group A 0 := C A ( [ V , A ] ) ∩ C A ( V / C V ( A ) ) A_0 :=C_A([V,A]) \cap C_A(V/C_V(A)) is trivial or not. We show that if A 0 A_0 is trivial, G G is a rank one group associated to a quadratic Jordan division algebra. If A 0 A_0 is not trivial (which is always the case if A A is not abelian), then A 0 A_0 defines a subgroup G 0 G_0 of G G acting quadratically on V V . We will call G 0 G_0 the quadratic kernel of G G . By a result of Timmesfeld we have G 0 ≅ S L 2 ( J , R ) G_0 \cong \mathrm {SL}_2(J,R) for a ring R R and a special quadratic Jordan division algebra J ⊆ R J \subseteq R . We show that J J is either a Jordan algebra contained in a commutative field or a Hermitian Jordan algebra. In the second case G G is the special unitary group of a pseudo-quadratic form π \pi of Witt index 1 1 , in the first case G G is the rank one group for a Freudenthal triple system. These results imply that if ( V , G ) (V,G) is a quadratic pair such that no two distinct root groups commute and c h a r V ≠ 2 , 3 \mathrm {char} V\ne 2,3 , then G G is a unitary group or an exceptional algebraic group.