   ISSN 1088-6850(online) ISSN 0002-9947(print)

Infinite nodal noncommutative Jordan algebras; differentiably simple algebras

Author: D. R. Scribner
Journal: Trans. Amer. Math. Soc. 156 (1971), 381-389
MSC: Primary 17.40
DOI: https://doi.org/10.1090/S0002-9947-1971-0274544-6
MathSciNet review: 0274544
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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.

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Keywords: Differentiably simple algebras, derivatives, polynomial algebras, power series algebras, nodal noncommutative Jordan algebras, infinite-dimensional noncommutative Jordan algebras, simple infinite Lie algebras
Article copyright: © Copyright 1971 American Mathematical Society