Nodal algebras of dimension 𝑝³

Abstract

An algebra A is in K if and only if there exist p, n, and P such that A+ = Bn where multiplication in A is defined by 1 » df dg 2 ,-,/_i dXi OX/ in terms of the (dot) product in Bn, and the [xt, Xj]=XiX/—XjXi are arbitrary except for the condition that at least one of these commutators be nonsingular. Every simple nodal noncommutative Jordan algebra is in K, but not all the algebras in K are simple [3]. Properties of K have been studied in [2], [3], [4], and in several other papers, but the ideal structure of the nonsimple algebras of K is unknown. In this paper we give a complete list of the ideals of a Lie-admissible algebra in K with three generators. Theorem. // A EK is Lie-admissible and of dimension ps, then there exist generators x, y, z such that A = Fl-\-F[x, y, z] (vector space direct sum) with [y, x] = 1 + c-xp-l-yp~1, c E A, (1) [z, x] = y'T~1-m(z), m(z) E P[l, z], and [z, y] = x*-l-n(z), n(z) E F[l, z].