## Infinite nodal noncommutative Jordan algebras; differentiably simple algebras

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- by D. R. Scribner
- Trans. Amer. Math. Soc.
**156**(1971), 381-389 - DOI: https://doi.org/10.1090/S0002-9947-1971-0274544-6
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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

*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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*Determination of the differentiably simple rings with a minimal ideal*(to be published).

## Bibliographic Information

- © Copyright 1971 American Mathematical Society
- 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