Acta Scientiarum Universitatis Pekiniensis (Naturalum)

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ON LIE ALGEBRAS WITH A CYCLIC NILPOTENT DERIVATION

Ting Shth-sun (Department of Mathematics and Mechanics)

ABSTRACT

A derivation of a Lie algebra is said to be cyclic nilpotent, if, as a linear transformation, it is cyclic nilpotent, 1. e, its index of nilpotency is equal to the dimension of the algebra. A Lie algebra with such a derivation is called a galgebra. In this paper, the structure of z-algebras over an arbitrary field of characteristic zero is investigated.

This paper can b2-divided into two parts. In the first part, the following two main theorems are proved. —

Theorem 3. A z-algebra of dimension greater than 3 is necessarily solvable.

Theorem 4. A z-algebru of dimension greater than 3is nilpotent if and only if it has a non-trivial centre.

In this part, as a corollary, we also get the following result. If LZ is a Lie algebra of dimension n, n>3, 7€L and adz is nilpotent, then (adx)""t=0.

In the second part, the structures of all non-nilpotent z-algebras are completely determined. It is proved that, all non-nilpotent z-algebras fall into four classes, they are mutually non-isomorphic. For each fixed dimension greater than 3, all the four classes except one contain only a finite number of algebras and their numbers are n—1, 2, 2 respectively.. Over an algebraically closed field, the exceptional class also contains a finite number of algebras and its number is Fin this case, m must be even. Examples are given to show that the restriction to

algebraically closed fields for this class is essantial for the finiteness of number.