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010 a| 2022022769;

020 a| 9781470451349q| (paperback);

020 z| 9781470470227q| (pdf);

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100 1 a| Grüninger, Matthias,d| 1976-?| UNAUTHORIZED;

245 10 a| Cubic action of a rank one grouph| [electronic resource] /c| Matthias Grüninger.;

260 a| Providence, RI :b| American Mathematical Society,c| [2022];

300 a| v, 141 pages ;c| 26 cm.;

490 0 a| Memoirs of the American Mathematical Society,x| 0065-9266 ;v| Volume 276, Number 1356 (fourth of 7 numbers);

504 a| Includes bibliographical references.;

506 a| Available only to authorized users.;

520 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"--c| Provided by publisher.;

538 a| Mode of access: World Wide Web;

650 0 a| Group theory.=| ^A17120;

650 7 a| Group theory and generalizations -- Structure and classification of infinite or finite groups -- Groups with a $BN$-pair; buildings.2| msc;

650 7 a| Geometry -- Finite geometry and special incidence structures -- Buildings and the geometry of diagrams.2| msc;

650 7 a| Group theory and generalizations -- Linear algebraic groups and related topics -- Linear algebraic groups over arbitrary fields.2| msc;

650 7 a| Nonassociative rings and algebras -- Jordan algebras (algebras, triples and pairs) -- Jordan structures associated with other structures.2| msc;

655 0 a| Electronic books.=| ^A491897;

856 40 z| Full text available from Ebook Central - Academic Completeu| https://ebookcentral.proquest.com/lib/eastcarolina/detail.action?docID=6939721;

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Cubic action of a rank one group / Matthias Grüninger.

Author/creator	Grüninger, Matthias, 1976-
Format	Electronic
Publication Info	Providence, RI : American Mathematical Society, [2022]
Description	v, 141 pages ; 26 cm.
Supplemental Content	Full text available from Ebook Central - Academic Complete
Subjects	Group theory.

Series	Memoirs of the American Mathematical Society, 0065-9266 ; Volume 276, Number 1356 (fourth of 7 numbers)
Abstract	"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"-- Provided by publisher.
Bibliography note	Includes bibliographical references.
Access restriction	Available only to authorized users.
Technical details	Mode of access: World Wide Web
Genre/form	Electronic books.
LCCN	2022022769
ISBN	9781470451349 (paperback)
ISBN	(pdf)

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Cubic action of a rank one group / Matthias Grüninger.

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