On cubic action of a rank one group

2011/06/12 by Grüninger, Matthias
#17C50 #20E42 #20G15 #51E24 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1106.2310

Abstract

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] ≠ 0. We have to distinguish whether the group A₀ :=C_A([V,A]) ∩ C_A(V/C_V(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 which acts 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 π 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≠ 2,3, then G is a unitary group or an exceptional algebraic group.

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