Classification of three-dimensional zeropotent algebras over the real number field

A nonassociative algebra is defined to be zeropotent if the square of any element is zero. In this paper, we give a complete classification of three-dimensional zeropotent algebras over the real number field up to isomorphism. By restricting the result to the subclass of Lie algebras, we can obtain...

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Bibliographic Details
Published in:Communications in algebra 2018-11, Vol.46 (11), p.4663-4681
Main Authors: Shirayanagi, Kiyoshi, Takahasi, Sin-Ei, Tsukada, Makoto, Kobayashi, Yuji
Format: Article
Language:English
Subjects:
Algebra
Anticommutative algebras
Classification
Isomorphism
Lie groups
nonassociative algebras
Primary 17A30
real Lie algebras
real zeropotent algebras
Secondary 17D99, 17B99
the Bianchi classification
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Summary:A nonassociative algebra is defined to be zeropotent if the square of any element is zero. In this paper, we give a complete classification of three-dimensional zeropotent algebras over the real number field up to isomorphism. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional real Lie algebras, which is in accordance with the Bianchi classification. Moreover, three-dimensional zeropotent algebras over a real closed field are classified in the same manner as those over the real number field.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2018.1448852