Cubic Action of a Rank One Group

• Record Nr.: UNINA9910960164903321
• Autore: Grüninger Matthias
• Pubbl/distr/stampa: Providence : , : American Mathematical Society, , 2022; ©2022
• ISBN: 9781470470227; 9781470451349
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (154 pages)
• Collana: Memoirs of the American Mathematical Society ; ; v.276
• Classificazione: 20E4251E2420G1517C50
• Disciplina: 512/.2; 512.2
• Soggetti: Group theory; Group theory and generalizations -- Structure and classification of infinite or finite groups -- Groups with a $BN$-pair; buildings; Geometry -- Finite geometry and special incidence structures -- Buildings and the geometry of diagrams; Group theory and generalizations -- Linear algebraic groups and related topics -- Linear algebraic groups over arbitrary fields; Nonassociative rings and algebras -- Jordan algebras (algebras, triples and pairs) -- Jordan structures associated with other structures
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Preliminaries -- 2.1. Moufang sets -- 2.2. Rank one groups -- 2.3. Some ring theory -- 2.4. Jordan algebras -- 2.5. Envelopes of special Jordan algebras -- 2.6. Quadratic spaces and Clifford Jordan algebras -- 2.7. Involutory sets and pseudo-quadratic forms -- 2.8. Cubic norm structures -- 2.9. Freudenthal triple systems -- 2.10. Structurable algebras -- 2.11. The Clifford algebra of a Freudenthal triple system -- Chapter 3. Cubic Action -- Chapter 4. Examples of cubic modules -- 4.1. Pseudo-quadratic spaces -- 4.2. Adjoint action -- 4.3. The Tits-Kantor-Koecher module -- 4.4. Quadratic pairs without commuting root subgroups -- 4.5. Elementary groups of Freudenthal triple systems -- 4.6. Connection with Moufang Quadrangles -- 4.7. Suzuki and Ree groups -- Chapter 5. The structure of a cubic module -- ; Chapter 6. Construction of irreducible submodules -- Chapter 7. Cubic rank one groups with trivial quadratic kernel -- Chapter 8. A characterisation of the adjoint module of \PSL₂( ) -- Chapter 9. Cubic rank one groups with non-trivial quadratic kernel -- Chapter 10. Cubic rank one groups with Hermitian quadratic kernel -- Chapter 11. Cubic rank one groups with commutative quadratic kernel -- Bibliography -- Back Cover.
• Sommario/riassunto: "We consider a rank one group G = A,B acting 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"--