Infinite nodal noncommutative Jordan algebras; differentiably simple algebras

Abstract

The first result is that any differentiably simple algebra of the form A = F 1 + R 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 = F 1 + R A = F1 + R , R an ideal of A + {A^ + } , then A + {A^ + } is a subalgebra of a power series algebra and multiplication in A is determined by certain elements c i j {c_{ij}} in A as in \[ f g = f ⋅ g + 1 2 ∑ ∂ f ∂ x i ⋅ ∂ g ∂ x j ⋅ c i j , fg = f \cdot g + \frac {1}{2}\sum {\frac {{\partial f}}{{\partial {x_i}}} \cdot \frac {{\partial g}}{{\partial {x_j}}} \cdot {c_{ij}},} \] where c i j = − c j i {c_{ij}} = - {c_{ji}} and “ ⋅ \cdot ” is the multiplication in A + {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 i j {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.