Cubic Action of a Rank One Group Paperback / softback
by Matthias Gruninger
Part of the Memoirs of the American Mathematical Society series
Paperback / softback
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Description
We 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.
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Available to Order - This title is available to order, with delivery expected within 2 weeks
- Format:Paperback / softback
- Pages:141 pages
- Publisher:American Mathematical Society
- Publication Date:30/06/2022
- Category:
- ISBN:9781470451349
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Information
-
Available to Order - This title is available to order, with delivery expected within 2 weeks
- Format:Paperback / softback
- Pages:141 pages
- Publisher:American Mathematical Society
- Publication Date:30/06/2022
- Category:
- ISBN:9781470451349