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 of a finite dimensional Lie algebra over a field of characteristic zero. In the case when is semisimple we show that exists and, when is algebraically closed, is equal to the dimension of the largest simple summand of . As a result we characterize central-simplicity: is central simple if and only if .

**[B]**Yu. A. Bahturin,*Identical relations in Lie algebras*, VNU Science Press, b.v., Utrecht, 1987. Translated from the Russian by Bahturin. MR**886063****[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 Young diagrams to P.I. algebras*, J. Algebra**82**(1983), no. 2, 559–567. MR**704771**, 10.1016/0021-8693(83)90167-9**[Jac]**Nathan Jacobson,*Lie algebras*, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR**0143793****[M]**S. P. Mishchenko,*Growth of varieties of Lie algebras*, Uspekhi Mat. Nauk**45**(1990), no. 6(276), 25–45, 189 (Russian); English transl., Russian Math. Surveys**45**(1990), no. 6, 27–52. MR**1101331**, 10.1070/RM1990v045n06ABEH002710**[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. Petrogradskiĭ,*Growth of polynilpotent varieties of Lie algebras, and rapidly increasing entire functions*, Mat. Sb.**188**(1997), no. 6, 119–138 (Russian, with Russian summary); English transl., Sb. Math.**188**(1997), no. 6, 913–931. MR**1479133**, 10.1070/SM1997v188n06ABEH000232**[Ra]**Yu. P. Razmyslov,*Identities of algebras and their representations*, Translations of Mathematical Monographs, vol. 138, American Mathematical Society, Providence, RI, 1994. Translated from the 1989 Russian original by A. M. Shtern. MR**1291603****[R1]**Amitai Regev,*Existence of identities in 𝐴⊗𝐵*, Israel J. Math.**11**(1972), 131–152. MR**0314893****[R2]**A. H. England,*Complex variable methods in elasticity*, Wiley—Interscience [A division of John Wiley & Sons, Ltd.], London-New York-Sydney, 1971. MR**0464824****[R3]**Amitai Regev,*The polynomial identities of matrices in characteristic zero*, Comm. Algebra**8**(1980), no. 15, 1417–1467. MR**584301**, 10.1080/00927878008822526**[R4]**Amitai Regev,*Asymptotic values for degrees associated with strips of Young diagrams*, Adv. in Math.**41**(1981), no. 2, 115–136. MR**625890**, 10.1016/0001-8708(81)90012-8**[V]**I. B. Voličenko,*Bases of a free Lie algebra modulo 𝑇-ideals*, Dokl. Akad. Nauk BSSR**24**(1980), no. 5, 400–403, 475 (Russian, with English summary). MR**572819****[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