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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Infinite nodal noncommutative Jordan algebras; differentiably simple algebras
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by D. R. Scribner PDF
Trans. Amer. Math. Soc. 156 (1971), 381-389 Request permission

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.
References
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Additional 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