A note on Lie-admissible nilalgebras

Abstract

It is shown that a finite dimensional, flexible, powerassociative, Lie-admissible algebra W over a field of characteristic 0 is a nilalgebra and there exists a Cartan subalgebra of %which is nil in W. Let W be a flexible algebra, that is, a nonassociative algebra satisfying the flexible law (xy)x = x(yx). The algebra 9is defined as the same vector space as 9t but with a multiplication given by [x, y] = xy yx. Then 9 is said to be Lie-admissible 9is a Lie algebra. If W is finite dimensional, we consider a Cartan subalgebra $ of 9-. Since Dh Rh L7, is a derivation of 9t for every h in $ and $ is the Fitting null component of 9for D($5), it follows that $ is a subalgebra of 9. If 9is a simple Lie algebra, it is shown that 91 is a nilalgebra ([3] and [4]). In case Wis simple, the structure of 91 has been studied in [2] by using a Cartan subalgebra of 9which is nil in 9t. In this note we give a condition that 9 be a nilalgebra in terms of a Cartan subalgebra of 9-. THEOREM. Suppose that 91 is a finite dimensional, flexible, powerassociative, Lie-admissible algebra over a field of characteristic 0. Then 9 is a nilalgebra and there exists a Cartan subalgebra of 9which is nil in W. PROOF. The only if part is obvious. Let $ be a Cartan subalgebra of 9which is nil in 9. Let $1 be the nil radical of W (the maximal nil ideal of 9). Suppose that 9 is not a nilalgebra. Then the quotient algebra W = W/91 satisfies the assumptions in the theorem and is semisimple. Since any flexible power-associative algebra of characteristic 0 is strictly power-associative, it follows from [4] that 9 has an identity T. Since the characteristic is 0, it also follows from [1, p. 379] that the homomorphic image 5 of $ is a Cartan subalgebra of the Lie algebra W-. But then 1 is in 5, and since t is nil in 9, this is a contradiction. Therefore 9 is a nilalgebra. Received by the editors February 24, 1971. AMS 1970 subject classifications. Primary 17A20, 17B05.