Infinite nodal noncommutative Jordan algebras; differentiably simple algebras
HTML articles powered by AMS MathViewer
 by D. R. Scribner PDF
 Trans. Amer. Math. Soc. 156 (1971), 381389 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

Richard E. Block, Determination of the differentiably simple rings with a minimal ideal (to be published).
 K. McCrimmon, Jordan algebras of degree $1$, Bull. Amer. Math. Soc. 70 (1964), 702. MR 164995, DOI 10.1090/S000299041964111733
 T. S. Ravisankar, A note on a theorem of Kokoris, Proc. Amer. Math. Soc. 21 (1969), 355–356. MR 238911, DOI 10.1090/S00029939196902389115
 R. D. Schafer, Nodal noncommutative Jordan algebras and simple Lie algebras of characteristic $p$, Trans. Amer. Math. Soc. 94 (1960), 310–326. MR 117262, DOI 10.1090/S0002994719600117262X
 Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New YorkLondon, 1966. MR 0210757
 D. R. Scribner, Lieadmissible, nodal, noncommutative Jordan algebras, Trans. Amer. Math. Soc. 154 (1971), 105–111. MR 314919, DOI 10.1090/S0002994719710314919X
 Robert Lee Wilson, Nonclassical simple Lie algebras, Bull. Amer. Math. Soc. 75 (1969), 987–991. MR 268236, DOI 10.1090/S000299041969123281
 Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.TorontoLondonNew York, 1960. MR 0120249, DOI 10.1007/9783662292440
Additional Information
 © Copyright 1971 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 156 (1971), 381389
 MSC: Primary 17.40
 DOI: https://doi.org/10.1090/S00029947197102745446
 MathSciNet review: 0274544