Nodal noncommutative Jordan algebras

A finite-dimensional power-associative algebra 91 is said to be nodal [6] if every element of 91 can be written as al + z where a e gf, 1 is the unity element of 91 and z is nilpotent and if the set of all nilpotent elements is not a subalgebra of 91.In [3; 4], Kokoris has shown that every simple nodal noncommutative Jordan algebra of characteristic p j= 2 has the form 91 = 51 + 5ft with 91+ = %[_x1,--,xn] for some n where the generators are all nilpotent of index p and the multiplication is associative.If / and g are two elements of 91 then the multiplication table of 91 is given by fg=fog + YZ Wo^ocl} where the circle product is the product in 9t+ and Cij = XiXj -XjXf.In [7] Schfer considers nodal noncommutative Jordan algebras defined by a skew-symmetric bilinear form (i.e., c^-eg) and those with two generators.All of these algebras are Lie-admissible (i.e., 91" is a Lie algebra).Schfer obtained the derivation algebras of these algebras defined by a skew-symmetric bilinear form.Here, we examine all simple nodal noncommutative Jordan algebras that are Lie-admissible over a field g of characteristic p # 2. First a set of generators is obtained having properties suitable for further study.This set of generators is then used to find the algebras fJ(9I) of derivatives of 91 and the algebras adj 9T and (adj 91")'.Schfer has shown that all of the simple Lie algebras defined by Block [1] can be realized as (adj 91")' for some 91 that is simple, nodal noncommutative Jordan and Lie-admissible.Hence we have obtained a somewhat different formulation of these algebras.The question remains whether all of these algebras, (adj 91")', are in the class defined by Block.It is our intention to investigate this question in a subsequent paper.2. We define the mapping Dy = D(y) by xDy = xy-yx.

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