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OverviewWe consider a rank one group G = A,Bacting 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 := CA([V,A]) ?CA(V/CV(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 acting 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 ? 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. Full Product DetailsAuthor: Matthias GruningerPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.281kg ISBN: 9781470451349ISBN 10: 1470451344 Pages: 141 Publication Date: 30 June 2022 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsReviewsAuthor InformationMatthias Gruninger, Justus-Liebig-Universitat Giessen, Germany. Tab Content 6Author Website:Countries AvailableAll regions |