On cubic action of a rank one group

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 A 0 := C A ([V, A]) ∩ 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 quadratic kernel of G. By a result of Timmesfeld we have G 0 ∼ = SL 2 (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 charV = 2, 3, then G is a unitary group or an exceptional algebraic group. ContentsChapter 1. Introduction Chapter 2. Preliminaries 2.1. Moufang sets 2.2. Rank one groups 2.3. Some ring theory 2.4. Quadratic Jordan algebras 2.5. Envelopes of special quadratic Jordan algebras 2.6. Involutory sets and pseudo-quadratic forms 2.7. Quadrangular algebras and Moufang quadrangles 2.8. Freudenthal triple systems and structurable algebras Chapter 3. Cubic Action Chapter 4. Examples for cubic modules 4.1. Pseudoquadratic spaces 4.2. Adjoint action 4.3. The Tits-Kantor-Koecher module 4.4. Quadratic pairs without commuting root groups 4.5. Quadratic spaces 4.6. Connection to Moufang Quadrangles 4.7. Suzuki and Ree groups Chapter 5. On the structure of a cubic module Chapter 6. Construction of irreducible submodules Chapter 7. Cubic rank one groups with trivial kernel Chapter 8. A characterisation of the adjoint module Chapter 9. Cubic rank one groups with non-trivial kernel Chapter 10. Cubic rank one groups with hermitian quadratic kernel Chapter 11. The construction of a pseudoquadratic space Chapter 12. Cubic rank one groups with commutative quadratic kernel Bibliography v CHAPTER 1

For a study of definable representations of algebraic groups

“…Proof. This will be tedious; the author strongly suspects that the second half of the argument could be made shorter with more culture on Jordan rings; incidently, it is not surprising to see Jordan rings appear in a cubic action (see the preprint [11] with the proviso that our setting smoothes many arguments).…”

Symmetric powers of Nat SL(2,𝕂)

“…The present work could be taken as an insane expansion of the former; see our final corollary. (Very parenthetically said, but still in length 3: our work is independent from the more general study led by M. Grüninger [5], which takes place in Timmesfeld's theory of abstract "rank one groups" [9]. Grüninger deals with abstract groups not necessarily isomorphic with SL 2 (K), and this loose assumption leads to numerous difficulties we ignore by being restrictive on the group.…”