LEADER 04911nam 22005413 450 001 9910915876803321 005 20231110214821.0 010 $a9781470470227$b(electronic bk.) 010 $z9781470451349 035 $a(MiAaPQ)EBC6939721 035 $a(Au-PeEL)EBL6939721 035 $a(CKB)21420567000041 035 $a(EXLCZ)9921420567000041 100 $a20220327d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCubic Action of a Rank One Group 210 1$aProvidence :$cAmerican Mathematical Society,$d2022. 210 4$d©2022. 215 $a1 online resource (154 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.276 311 08$aPrint version: Grüninger, Matthias Cubic Action of a Rank One Group Providence : American Mathematical Society,c2022 9781470451349 327 $aCover -- 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 \PSL?( ) -- 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. 330 $a"We consider a rank one group G = A,B acting cubically on a module V , this means [V, A, A,A] = 0 but [V, G, G,G] = 0. We have to distinguish whether the group A0 := CA([V,A]) CA(V/CV (A)) is trivial or not. We show that if A0 is trivial, G is a rank one group associated to a quadratic Jordan division algebra. If A0 is not trivial (which is always the case if A is not abelian), then A0 defines a subgroup G0 of G acting quadratically on V . We will call G0 the quadratic kernel of G. By a result of Timmesfeld we have G0 = SL2(J,R) for a ring R and a special quadratic Jordan division algebra J R. We show that J is either a Jordan algebra contained in a commutative field or a Hermitian Jordan algebra. In the second case G is the special unitary group of a pseudo-quadratic form of Witt index 1, in the first case G is the rank one group for a Freudenthal triple system. These results imply that if (V,G) is a quadratic pair such that no two distinct root groups commute and charV = 2, 3, then G is a unitary group or an exceptional algebraic group"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aGroup theory 606 $aGroup theory and generalizations -- Structure and classification of infinite or finite groups -- Groups with a $BN$-pair; buildings$2msc 606 $aGeometry -- Finite geometry and special incidence structures -- Buildings and the geometry of diagrams$2msc 606 $aGroup theory and generalizations -- Linear algebraic groups and related topics -- Linear algebraic groups over arbitrary fields$2msc 606 $aNonassociative rings and algebras -- Jordan algebras (algebras, triples and pairs) -- Jordan structures associated with other structures$2msc 615 0$aGroup theory. 615 7$aGroup theory and generalizations -- Structure and classification of infinite or finite groups -- Groups with a $BN$-pair; buildings. 615 7$aGeometry -- Finite geometry and special incidence structures -- Buildings and the geometry of diagrams. 615 7$aGroup theory and generalizations -- Linear algebraic groups and related topics -- Linear algebraic groups over arbitrary fields. 615 7$aNonassociative rings and algebras -- Jordan algebras (algebras, triples and pairs) -- Jordan structures associated with other structures. 676 $a512/.2 676 $a512.2 686 $a20E42$a51E24$a20G15$a17C50$2msc 700 $aGrüninger$b Matthias$01778644 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a9910915876803321 996 $aCubic Action of a Rank One Group$94301572 997 $aUNINA