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30) (k - j)ej+k - 31 - The resulting Lie algebra is called the where we have set V~o~o algebka, which plays a crucial role in dual strings theory. 29) is the ]aQObhon-Witt algebka (Jacobson [l]). m 1 e 0 ,•••,e0 form a basis of an abelian We note that the elements H . a root space for [e i ej] = a ej i a 0' a b Y e 1 ••• em W and the linear span of H of Cartan subalgebra (1 29) ' • determines W is not classical. 14. be realized as a Z-graded subalgebra of the derivation algebra of an affine Kac-Moody algebra with one-dimensional center.

26. characteristic ~ 2 dim A 4 , then ~ Let A be a flexible nilalgebra over a field such that A- F is a nilpotent Malcev algebra. of If A is also nilpotent such that all products of any 4 elements in A are zero. Proof. dim A ~ dim A We first treat the case where 2 it is easily seen that 3 Thus the nil-index of x, x2 , x3 A is spanned by hence A is associative and A is less than 5. for some element A4 = 0 A is associative such that A2 = 0 then since dim A = 3 or 4 A3 = 0 . and the nil-index of for all x,y,z (1.

Such that Assume that is either isomorphic to a classical Lie algebra of A characteristic 2,3, or to a Kac-Moody algebra without center. ;t Cartan subalgebra of A algebra ismorphic to A- - When A is a nil subalgebra of has center the product of I and J let a since A 0 J I A or a Cartan subalgebra of and b E J with is an ideal of I + J maximal nil ideal A I + then am : 0 is nil. is nil under F (a + b)mn ~~Qat . is nilpotent. te. Let A be a finite-dimensional flexible power- associative Malcev-admissible algebra over A- is semisimple.