Infinite Nodal Noncommutative Jordan Algebras; Differentiably Simple Algebras

Abstract

The first result is that any differentiably simple algebra of the form $A = F1 + R$, for R a proper ideal, 1 the identity element, and F the base field, must be a subalgebra of a (commutative associative) power series algebra over F, and is truncated if the characteristic is not zero. Moreover the algebra A contains the polynomial subalgebra generated by the indeterminates and identity of the power series algebra. This is used to prove that if A is any simple flexible algebra of the form $A = F1 + R$, R an ideal of ${A^ + }$, then ${A^ + }$ is a subalgebra of a power series algebra and multiplication in A is determined by certain elements ${c_{ij}}$ in A as in \[ fg = f \cdot g + \frac {1}{2}\sum {\frac {{\partial f}}{{\partial {x_i}}} \cdot \frac {{\partial g}}{{\partial {x_j}}} \cdot {c_{ij}},} \] where ${c_{ij}} = - {c_{ji}}$ and “$\cdot$” is the multiplication in ${A^ + }$. This applies in particular to simple nodal noncommutative Jordan algebras (of characteristic not 2). These results suggest a method of constructing noncommutative Jordan algebras of the given form. We have done this with the restriction that the ${c_{ij}}$ lie in F1. The last result is that if A is a finitely generated simple noncommutative algebra of characteristic 0 of this form, then Der (A) is an infinite simple Lie algebra of a known type.