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Transactions of the American Mathematical Society
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
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 $\dim B = \lim _{n\to \infty } \sqrt[{n}]{c_ n^{L} (B)}$.


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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: http://dx.doi.org/10.1090/S0002-9947-99-02419-8
PII: S 0002-9947(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