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Transactions of the American Mathematical Society

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

 
 

 

Simple and semisimple Lie algebras
and codimension growth


Authors: Antonio Giambruno, Amitai Regev and Michail V. Zaicev
Journal: Trans. Amer. Math. Soc. 352 (2000), 1935-1946
MSC (2000): Primary 17B01, 17B20, 16R10; Secondary 20C30, 17C05
DOI: https://doi.org/10.1090/S0002-9947-99-02419-8
Published electronically: December 14, 1999
MathSciNet review: 1637070
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the exponential growth of the codimensions $c_ n^{L}(B)$ of a finite dimensional Lie algebra $B$ over a field of characteristic zero. In the case when $B$ is semisimple we show that $\lim _{n\to \infty } \sqrt[{n}]{c_ n^{L}(B)}$ exists and, when $F$ is algebraically closed, is equal to the dimension of the largest simple summand of $B$. As a result we characterize central-simplicity: $B$ is central simple if and only if $\dim B = \lim _{n\to \infty } \sqrt[{n}]{c_ n^{L} (B)}$.


References [Enhancements On Off] (What's this?)

  • [B] Yu. A. Bahturin, Identical Relations in Lie Algebras, VNU Science Press, Utrecht, 1987. MR 88f:17032
  • [BMR] Yu. A. Bahturin, S. P. Mischenko and A. Regev, On the Lie and associative codimension growth, preprint.
  • [BR] A. Berele and A. Regev, Applications of hook diagrams to P.I. algebras, J. Algebra 82 (1983), 559-567. MR 84g:16012
  • [Jac] N. Jacobson, Lie Algebras, Interscience Publishers, 1962. MR 26:1345
  • [M] S. P. Mischenko, Growth of varieties of Lie algebras, Uspekhi Matem. Nauk. 45 (1990), 25-45; English transl., Russian Math. Surveys 45 (1990), no. 6, 27-52. MR 92g:17003
  • [MP] S. P. Mischenko and V. M. Petrogradsky, Exponents of varieties of Lie algebras with a nilpotent commutator subalgebra, Comm. Algebra 27 (1999), 2223-2230. CMP 99:11
  • [P] V. M. Petrogradsky, Growth of polynilpotent varieties of Lie algebras and fast increasing entire functions, Mat. Sb. 188 (1997), no. 6, 119-138; English transl., Sb. Math. 188 (1997), 913-931. MR 99a:17008
  • [Ra] Yu. P. Razmyslov, Identities of algebras and their representations, Transl. Math. Monogr. vol. 138, Amer. Math. Soc., Providence, RI, 1994. MR 95i:16022
  • [R1] A. Regev, Existence of identities in $A\otimes B$, Israel J. Math 11 (1972), 131-152. MR 47:3442
  • [R2] A. Regev, The representations of $S\sb n$ and explicit identities for P.I. algebras, J. Algebra 51 (1978), 25-40. MR 57:4745
  • [R3] A. Regev, The polynomial identities of matrices in characteristic zero, Comm. Algebra 8 (1980), 1417-1467. MR 83c:15015
  • [R4] A. Regev, Asymptotic values for degrees associated with stripes of Young diagrams, Adv. Math. 41 (1981), 115-136. MR 82h:20015
  • [V] I. B. Volichenko, Bases of a free Lie algebra modulo $T$-ideals, Dokl. Akad. Nauk BSSR 24 (1980), n. 5, 400-403. (Russian) MR 81h:17007
  • [ZM] M. V. Zaicev and S. P. Mishchenko, Varieties of Lie subalgebras of polynomial growth, Uspekhi Matem. Nauk 52 (1997), no. 2, 165-166; English transl., Russian Math. Surveys 52 (1997), 432-433. CMP 98:04
  • [Z] M. V. Zaicev, Identities of affine Kac-Moody algebras, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1996, no. 2, 33-36; English transl., Moscow Univ. Math. Bull. 51 (1996), no. 2, 29-31. CMP 98:05

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Additional Information

Antonio Giambruno
Affiliation: Department of Mathematics, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy
Email: a.giambruno@unipa.it

Amitai Regev
Affiliation: Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel and The Pennsylvania State University, University Park, Pennsylvania 16802
Email: regev@wisdom.weizmann.ac.il, regev@math.psu.edu

Michail V. Zaicev
Affiliation: Department of Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia
Email: zaicev@nw.math.msu.su

DOI: https://doi.org/10.1090/S0002-9947-99-02419-8
Keywords: Lie algebras, polynomial identities, codimensions
Received by editor(s): November 27, 1997
Published electronically: December 14, 1999
Additional Notes: The first author was partially supported by MURST and CNR of Italy
The second author was partially supported by NSF Grant No. DMS-94-01197
The third author was partially supported by RFFI grants 96-01-00146 and 96-15-96050
Article copyright: © Copyright 2000 American Mathematical Society

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