Classification of three-dimensional zeropotent algebras over an algebraically closed field

ABSTRACT

A nonassociative algebra is defined to be zeropotent if the square of any element is zero. Zeropotent algebras are exactly the same as anticommutative algebras when the characteristic of the ground field is not two. The class of zeropotent algebras properly contains that of Lie algebras. In this paper, we give a complete classification of three-dimensional zeropotent algebras over an algebraically closed field of characteristic not equal to two. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional complex Lie algebras, which is in accordance with the conventional one.

KEYWORDS:

• Anticommutative algebras
• Jacobi element
• Lie algebras
• nonassociative algebras
• zeropotent algebras

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 17A30
• Secondary 17D99, 17B99

Notes

¹The terminology “zeropotent” is also used for the more general algebraic structure groupoid in [4].

²We found such a matrix by solving the system of nine (or less) algebraic equations in nine variables derived from a given matrix equation ${}^{t}XAX=A^{\prime }$. In general, however, this task is very hard and so we often used some computational algebraic techniques based on the theory of Gröbner bases. This is not the subject of the present paper and we will not go into the details. Most transformation matrices that appear in this paper were computed in this manner. In some of the computations, we used computer algebra system Mathematica or Maple, especially for Gröbner basis computation and linear algebra operations. However, we finally checked that every transformation matrix so obtained satisfies the associated matrix equation by hand calculations.