Constructing bases for radicals and nilradicals of Lie algebras
Author:
Stephen Merrin
Journal:
Proc. Amer. Math. Soc. 119 (1993), 681690
MSC:
Primary 17B05; Secondary 03F65
MathSciNet review:
1152285
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Abstract: The radical and nilradical of a finitedimensional Lie algebra are well defined unique subspaces of . Nevertheless, we show the impossibility of ever finding a general algorithm that will construct finite bases for radicals (or nilradicals) of arbitrary finitedimensional 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 finitedimensional Lie algebras over a discrete field can always be constructed if and only if 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).
 [1]
Robert
E. Beck, Bernard
Kolman, and Ian
N. Stewart, Computing the structure of a Lie algebra,
Computers in nonassociative rings and algebras (Special session, 82nd
Annual Meeting Amer. Math. Soc., San Antonio, Tex., 1976), Academic Press,
New York, 1977, pp. 167–188. MR 0486008
(58 #5800)
 [2]
Errett
Bishop, Foundations of constructive analysis, McGrawHill Book
Co., New York, 1967. MR 0221878
(36 #4930)
 [3]
Nathan
Jacobson, Lie algebras, Dover Publications Inc., New York,
1979. Republication of the 1962 original. MR 559927
(80k:17001)
 [4]
S. Merrin, A strong constructive version of Engel's theorem, submitted.
 [5]
, Some constructive results in the theory of Lie algebras, Ph.D. dissertation, New Mexico State Univ., 1990.
 [6]
Ray
Mines, Fred
Richman, and Wim
Ruitenburg, A course in constructive algebra, Universitext,
SpringerVerlag, New York, 1988. MR 919949
(89d:03066)
 [7]
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 0294327
(45 #3396)
 [1]
 R. Beck, B. Kolman, and I. Stewart, Computing the structure of a Lie algebra, Computers in Nonassociative Rings and Algebras (R. Beck and B. Kolman, eds.), Academic Press, New York, 1977, pp. 167188. MR 0486008 (58:5800)
 [2]
 E. Bishop, Foundations of constructive analysis, McGrawHill, New York, 1967. MR 0221878 (36:4930)
 [3]
 N. Jacobson, Lie algebras, Dover, New York, 1979. MR 559927 (80k:17001)
 [4]
 S. Merrin, A strong constructive version of Engel's theorem, submitted.
 [5]
 , Some constructive results in the theory of Lie algebras, Ph.D. dissertation, New Mexico State Univ., 1990.
 [6]
 R. Mines, F. Richman, and W. Ruitenburg, A course in constructive algebra, SpringerVerlag, New York, 1988. MR 919949 (89d:03066)
 [7]
 A. Seidenberg, Construction of the integral closure of a finite integral domain, Rend. Sem. Mat. Fis. Milano 40 (1970), 100120. MR 0294327 (45:3396)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199311522850
PII:
S 00029939(1993)11522850
Keywords:
Lie algebra,
constructive algebra,
radical,
nilradical
Article copyright:
© Copyright 1993 American Mathematical Society
