Jordan centroids
作者:
Kevin McCrimmon,
期刊:
Communications in Algebra
(Taylor Available online 1999)
卷期:
Volume 27,
issue 2
页码: 933-954
ISSN:0092-7872
年代: 1999
DOI:10.1080/00927879908826470
出版商: Gordon and Breach Science Publishers Ltd.
数据来源: Taylor
摘要:
Central simple triples are important for the classification of prime Jordan triples of Clifford type in arbitrary characterstics. For triples and pairs (or even for unital Jordan algebras of characteristic 2), there is no workable notion of center, and the concept of “central simple” system must be understood as “centroid-simple”. The centroid of a Jordan system (algebra, triple, or pair) consists of the “natural” scalars for that system: the largest unital, commutative ring Γ such that the system can be considered as a quadratic Jordan system over Γ. In this paper we will characterize the centroids of the basic simple Jordan algebras, triples, and pairs. (Consideration of the tangled ample outer ideals in Jordan algebras of quadratic forms will be left to a separate paper.) A powerful tool is the Eigenvalue Lemma, that a centroidal transformation on a prime system over φ which has an eigenvalue α in φ must actually be scalar multiplication by α. An important consequence is that a prime system over φ with reduced elementsPxJ= φx (or which grows reduced elements under controlled conditions) must already be central, Γ = φ.
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