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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Constructing bases for radicals and nilradicals of Lie algebras
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by Stephen Merrin PDF
Proc. Amer. Math. Soc. 119 (1993), 681-690 Request permission

Abstract:

The radical and nilradical of a finite-dimensional Lie algebra $L$ are well defined unique subspaces of $L$. Nevertheless, we show the impossibility of ever finding a general algorithm that will construct finite bases for radicals (or nilradicals) of arbitrary finite-dimensional Lie algebras. Our approach involves an investigation of the relationship between radicals of associative algebras and radicals of Lie algebras. Building on a result of Richman in the constructive theory of associative algebras, we prove that bases for radicals and nilradicals of finite-dimensional Lie algebras over a discrete field $F$ can always be constructed if and only if $F$ satisfies Seidenberg’s condition P. A special case is that if we restrict ourselves to fields of characteristic zero, we can indeed always construct bases for radicals. Our proofs are entirely constructive (i.e., do not use the general law of excluded middle).
References
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  • Errett Bishop, Foundations of constructive analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0221878
  • Nathan Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR 559927
    • S. Merrin, A strong constructive version of Engel’s theorem, submitted. —, Some constructive results in the theory of Lie algebras, Ph.D. dissertation, New Mexico State Univ., 1990.
  • Ray Mines, Fred Richman, and Wim Ruitenburg, A course in constructive algebra, Universitext, Springer-Verlag, New York, 1988. MR 919949, DOI 10.1007/978-1-4419-8640-5
  • Abraham Seidenberg, Construction of the integral closure of a finite integral domain, Rend. Sem. Mat. Fis. Milano 40 (1970), 100–120 (English, with Italian summary). MR 294327, DOI 10.1007/BF02923228
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 119 (1993), 681-690
  • MSC: Primary 17B05; Secondary 03F65
  • DOI: https://doi.org/10.1090/S0002-9939-1993-1152285-0
  • MathSciNet review: 1152285