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 , 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 , *R* an ideal of , then is a subalgebra of a power series algebra and multiplication in *A* is determined by certain elements in *A* as in

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 lie in *F*1. 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0274544-6

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