On the codimension growth of -graded algebras

Author:
Eli Aljadeff

Journal:
Proc. Amer. Math. Soc. **138** (2010), 2311-2320

MSC (2010):
Primary 16P90, 16R10, 16W50

Published electronically:
March 10, 2010

MathSciNet review:
2607860

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Abstract: Let be an associative *PI*-affine algebra over a field of characteristic zero. Suppose is -graded where is a finite group. Let and denote the codimension growth of and of the identity component , respectively. We prove This inequality had been conjectured by Bahturin and Zaicev.

**1.**E. Aljadeff, A. Giambruno and D. La Mattina,*Graded polynomial identities and exponential growth*, to appear in J. Reine Angew. Math., arXiv:0903.1860.**2.**E. Aljadeff and A. Kanel-Belov,*Representability and Specht problem for -graded algebras*, arXiv:0903.0362.**3.**Yu. A. Bakhturin, M. V. Zaĭtsev, and S. K. Segal,*Finite-dimensional simple graded algebras*, Mat. Sb.**199**(2008), no. 7, 21–40 (Russian, with Russian summary); English transl., Sb. Math.**199**(2008), no. 7-8, 965–983 (2008). MR**2488221**, 10.1070/SM2008v199n07ABEH003949**4.**Yu. A. Bahturin and M. V. Zaicev,*Identities of graded algebras and codimension growth*, Trans. Amer. Math. Soc.**356**(2004), no. 10, 3939–3950 (electronic). MR**2058512**, 10.1090/S0002-9947-04-03426-9**5.**Francesca Benanti, Antonio Giambruno, and Manuela Pipitone,*Polynomial identities on superalgebras and exponential growth*, J. Algebra**269**(2003), no. 2, 422–438. MR**2015285**, 10.1016/S0021-8693(03)00528-3**6.**M. Cohen and S. Montgomery,*Group-graded rings, smash products, and group actions*, Trans. Amer. Math. Soc.**282**(1984), no. 1, 237–258. MR**728711**, 10.1090/S0002-9947-1984-0728711-4**7.**A. Giambruno, S. Mishchenko, and M. Zaicev,*Group actions and asymptotic behavior of graded polynomial identities*, J. London Math. Soc. (2)**66**(2002), no. 2, 295–312. MR**1920403**, 10.1112/S0024610702003435**8.**A. Giambruno, S. Mishchenko, and M. Zaicev,*Codimensions of algebras and growth functions*, Adv. Math.**217**(2008), no. 3, 1027–1052. MR**2383893**, 10.1016/j.aim.2007.07.008**9.**A. Giambruno and M. Zaicev,*On codimension growth of finitely generated associative algebras*, Adv. Math.**140**(1998), no. 2, 145–155. MR**1658530**, 10.1006/aima.1998.1766**10.**A. Giambruno and M. Zaicev,*Exponential codimension growth of PI algebras: an exact estimate*, Adv. Math.**142**(1999), no. 2, 221–243. MR**1680198**, 10.1006/aima.1998.1790**11.**Antonio Giambruno and Mikhail Zaicev,*Polynomial identities and asymptotic methods*, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR**2176105****12.**Aleksandr Robertovich Kemer,*Ideals of identities of associative algebras*, Translations of Mathematical Monographs, vol. 87, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by C. W. Kohls. MR**1108620****13.**A. R. Kemer,*Identities of finitely generated algebras over an infinite field*, Izv. Akad. Nauk SSSR Ser. Mat.**54**(1990), no. 4, 726–753 (Russian); English transl., Math. USSR-Izv.**37**(1991), no. 1, 69–96. MR**1073084****14.**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**15.**D. Ştefan and F. Van Oystaeyen,*The Wedderburn-Malcev theorem for comodule algebras*, Comm. Algebra**27**(1999), no. 8, 3569–3581. MR**1699590**, 10.1080/00927879908826648**16.**M. V. Zaĭtsev,*Integrality of exponents of growth of identities of finite-dimensional Lie algebras*, Izv. Ross. Akad. Nauk Ser. Mat.**66**(2002), no. 3, 23–48 (Russian, with Russian summary); English transl., Izv. Math.**66**(2002), no. 3, 463–487. MR**1921808**, 10.1070/IM2002v066n03ABEH000386

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Additional Information

**Eli Aljadeff**

Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel

Email:
aljadeff@tx.technion.ac.il

DOI:
http://dx.doi.org/10.1090/S0002-9939-10-10282-2

Keywords:
Graded algebra,
polynomial identity

Received by editor(s):
August 29, 2009

Received by editor(s) in revised form:
November 3, 2009

Published electronically:
March 10, 2010

Additional Notes:
The author was partially supported by the Israel Science Foundation (grant No. 1283/08) and by the E. Schaver Research Fund

Communicated by:
Martin Lorenz

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.