On the codimension growth of $G$-graded algebras
HTML articles powered by AMS MathViewer
- by Eli Aljadeff PDF
- Proc. Amer. Math. Soc. 138 (2010), 2311-2320 Request permission
Abstract:
Let $W$ be an associative PI-affine algebra over a field $F$ of characteristic zero. Suppose $W$ is $G$-graded where $G$ is a finite group. Let $\exp (W)$ and $\exp (W_{e})$ denote the codimension growth of $W$ and of the identity component $W_{e}$, respectively. We prove $\exp (W)\leq |G|^2 \exp (W_{e}).$ This inequality had been conjectured by Bahturin and Zaicev.References
- E. Aljadeff, A. Giambruno and D. La Mattina, Graded polynomial identities and exponential growth, to appear in J. Reine Angew. Math., arXiv:0903.1860.
- E. Aljadeff and A. Kanel-Belov, Representability and Specht problem for $G$-graded algebras, arXiv:0903.0362.
- 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, DOI 10.1070/SM2008v199n07ABEH003949
- Yu. A. Bahturin and M. V. Zaicev, Identities of graded algebras and codimension growth, Trans. Amer. Math. Soc. 356 (2004), no. 10, 3939–3950. MR 2058512, DOI 10.1090/S0002-9947-04-03426-9
- Francesca Benanti, Antonio Giambruno, and Manuela Pipitone, Polynomial identities on superalgebras and exponential growth, J. Algebra 269 (2003), no. 2, 422–438. MR 2015285, DOI 10.1016/S0021-8693(03)00528-3
- 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, DOI 10.1090/S0002-9947-1984-0728711-4
- 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, DOI 10.1112/S0024610702003435
- A. Giambruno, S. Mishchenko, and M. Zaicev, Codimensions of algebras and growth functions, Adv. Math. 217 (2008), no. 3, 1027–1052. MR 2383893, DOI 10.1016/j.aim.2007.07.008
- A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), no. 2, 145–155. MR 1658530, DOI 10.1006/aima.1998.1766
- A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142 (1999), no. 2, 221–243. MR 1680198, DOI 10.1006/aima.1998.1790
- Antonio Giambruno and Mikhail Zaicev, Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR 2176105, DOI 10.1090/surv/122
- 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, DOI 10.1090/mmono/087
- 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, DOI 10.1070/IM1991v037n01ABEH002053
- Amitai Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math. 41 (1981), no. 2, 115–136. MR 625890, DOI 10.1016/0001-8708(81)90012-8
- D. Ştefan and F. Van Oystaeyen, The Wedderburn-Malcev theorem for comodule algebras, Comm. Algebra 27 (1999), no. 8, 3569–3581. MR 1699590, DOI 10.1080/00927879908826648
- 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, DOI 10.1070/IM2002v066n03ABEH000386
Additional Information
- Eli Aljadeff
- Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
- MR Author ID: 229998
- Email: aljadeff@tx.technion.ac.il
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 2311-2320
- MSC (2010): Primary 16P90, 16R10, 16W50
- DOI: https://doi.org/10.1090/S0002-9939-10-10282-2
- MathSciNet review: 2607860