Infinite nodal noncommutative Jordan algebras; differentiably simple algebras
Author:
D. R. Scribner
Journal:
Trans. Amer. Math. Soc. 156 (1971), 381389
MSC:
Primary 17.40
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 where and ``'' is the multiplication 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 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.
 [1]
Richard E. Block, Determination of the differentiably simple rings with a minimal ideal (to be published).
 [2]
K.
McCrimmon, Jordan algebras of degree 1,
Bull. Amer. Math. Soc. 70 (1964), 702. MR 0164995
(29 #2286), http://dx.doi.org/10.1090/S000299041964111733
 [3]
T.
S. Ravisankar, A note on a theorem of
Kokoris, Proc. Amer. Math. Soc. 21 (1969), 355–356. MR 0238911
(39 #271), http://dx.doi.org/10.1090/S00029939196902389115
 [4]
R.
D. Schafer, Nodal noncommutative Jordan algebras
and simple Lie algebras of characteristic 𝑝, Trans. Amer. Math. Soc. 94 (1960), 310–326. MR 0117262
(22 #8044), http://dx.doi.org/10.1090/S0002994719600117262X
 [5]
Richard
D. Schafer, An introduction to nonassociative algebras, Pure
and Applied Mathematics, Vol. 22, Academic Press, New YorkLondon, 1966. MR 0210757
(35 #1643)
 [6]
D.
R. Scribner, Lieadmissible, nodal, noncommutative
Jordan algebras, Trans. Amer. Math. Soc. 154 (1971), 105–111.
MR
0314919 (47 #3468), http://dx.doi.org/10.1090/S0002994719710314919X
 [7]
Robert
Lee Wilson, Nonclassical simple Lie
algebras, Bull. Amer. Math. Soc. 75 (1969), 987–991. MR 0268236
(42 #3135), http://dx.doi.org/10.1090/S000299041969123281
 [8]
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
(22 #11006)
 [1]
 Richard E. Block, Determination of the differentiably simple rings with a minimal ideal (to be published).
 [2]
 Kevin McCrimmon, Jordan algebras of degree 1, Bull. Amer. Math. Soc. 70 (1964), 702. MR 29 #2286. MR 0164995 (29:2286)
 [3]
 T. S. Ravisankar, A note on a theorem of Kokoris, Proc. Amer. Math. Soc. 21 (1969), 355356. MR 39 #271. MR 0238911 (39:271)
 [4]
 R. D. Schafer, Nodal noncommutative Jordan algebras and simple Lie algebras of characteristic p, Trans. Amer. Math. Soc. 94 (1960), 310326. MR 22 #8044. MR 0117262 (22:8044)
 [5]
 , An introduction to nonassociative algebras, Pure and Appl. Math., vol. 22, Academic Press, New York, 1966. MR 35 #1643. MR 0210757 (35:1643)
 [6]
 D. R. Scribner, Lieadmissible, nodal, noncommutative Jordan algebras, Trans. Amer. Math. Soc. 154 (1971), 105111. MR 0314919 (47:3468)
 [7]
 R. L. Wilson, Nonclassical simple Lie algebras, Bull. Amer. Math. Soc. 75 (1969), 987991. MR 0268236 (42:3135)
 [8]
 Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. 2, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #11006. MR 0120249 (22:11006)
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DOI:
http://dx.doi.org/10.1090/S00029947197102745446
PII:
S 00029947(1971)02745446
Keywords:
Differentiably simple algebras,
derivatives,
polynomial algebras,
power series algebras,
nodal noncommutative Jordan algebras,
infinitedimensional noncommutative Jordan algebras,
simple infinite Lie algebras
Article copyright:
© Copyright 1971
American Mathematical Society
