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

ABSTRACT

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.

KEYWORDS:

• Anticommutative algebras
• nonassociative algebras
• real Lie algebras
• real zeropotent algebras
• the Bianchi classification

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 17A30
• Secondary 17D99, 17B99