American Institute of Mathematical Sciences

Advanced
  2014, 21: 113-119. doi: 10.3934/era.2014.21.113

On existence of PI-exponents of codimension growth

Mikhail Zaicev 1,

1. 

Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992, Russian Federation

Received  January 2014 Revised  March 2014 Published  June 2014

We construct a family of examples of non-associative algebras $\{R_\alpha \,\vert\, 1<\alpha\in\mathbb R\}$ such that $\underline{\exp}(R_\alpha)=1$, $\overline{\exp}(R_\alpha)=\alpha$. In particular, it follows that for any $R_\alpha$, an ordinary PI-exponent of codimension growth does not exist.
Keywords: non-associative algebra, PI-exponent., Polynomial identity, codimensions, exponential growth.
Mathematics Subject Classification: Primary: 16R10; Secondary: 16P9.
Citation: Mikhail Zaicev. On existence of PI-exponents of codimension growth. Electronic Research Announcements, 2014, 21: 113-119. doi: 10.3934/era.2014.21.113
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show all references

References:
[1]

Translated from the Russian by Bahturin, VNU Science Press, b.v., Utrecht, 1987.  Google Scholar

[2]

Linear Algebra Appl., 357 (2002), 15-34. doi: 10.1016/S0024-3795(02)00356-7.  Google Scholar

[3]

Israel J. Math., 156 (2006), 73-91. doi: 10.1007/BF02773825.  Google Scholar

[4]

Adv. in Appl. Math., 41 (2008), 52-75. doi: 10.1016/j.aam.2007.03.002.  Google Scholar

[5]

Comm. Algebra, 38 (2010), 3506-3510. doi: 10.1080/00927870902939426.  Google Scholar

[6]

Israel J. Math., 103 (1998), 17-28. doi: 10.1007/BF02762265.  Google Scholar

[7]

Graduate Course in Algebra, Springer-Verlag Singapore, Singapore, 2000.  Google Scholar

[8]

Comm. Algebra, 39 (2011), 2943-2952. doi: 10.1080/00927870903386494.  Google Scholar

[9]

Adv. in Appl. Math., 47 (2011), 125-139. doi: 10.1016/j.aam.2010.04.007.  Google Scholar

[10]

Adv. Math., 142 (1999), 221-243. doi: 10.1006/aima.1998.1790.  Google Scholar

[11]

Mathematical Surveys and Monographs, 122, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/122.  Google Scholar

[12]

Trans. Amer. Math. Soc., 362 (2010), 3107-3123. doi: 10.1090/S0002-9947-09-04865-X.  Google Scholar

[13]

J. Lond. Math. Soc. (2), 85 (2012), 534-548. doi: 10.1112/jlms/jdr059.  Google Scholar

[14]

(Russian) Sibirsk. Mat. Ž., 19 (1978), 54-69, 237.  Google Scholar

[15]

Trans. Amer. Math. Soc., 181 (1973), 429-438.  Google Scholar

[16]

(Russian) Uspehi Mat. Nauk, 27 (1972), 213-214.  Google Scholar

[17]

S. P. Mishchenko, Varieties of Lie algebras with weak growth of the sequence of codimensions,, (Russian) \emph{Vestnik Moskov. Univ. Ser. I Mat. Mekh.}, 1982 (): 63.   Google Scholar

[18]

(Russian) Uspekhi Mat. Nauk, 45 (1990), 25-45, 189; translation in Russian Math. Surveys, 45 (1990), 27-52. doi: 10.1070/RM1990v045n06ABEH002710.  Google Scholar

[19]

Comm. Algebra, 27 (1999), 2223-2230. doi: 10.1080/00927879908826560.  Google Scholar

[20]

Trans. Amer. Math. Soc., 359 (2007), 4669-4694. doi: 10.1090/S0002-9947-07-04008-1.  Google Scholar

[21]

J. Pure Appl. Algebra, 202 (2005), 82-101. doi: 10.1016/j.jpaa.2005.01.013.  Google Scholar

[22]

J. Lie Theory, 22 (2012), 465-479.  Google Scholar

[23]

Israel J. Math., 11 (1972), 131-152. doi: 10.1007/BF02762615.  Google Scholar

[24]

Israel J. Math., 47 (1984), 246-250. doi: 10.1007/BF02760520.  Google Scholar

[25]

(Russian) Sibirsk. Mat. Zh., 25 (1984), 40-54.  Google Scholar

[26]

in Methods in Ring Theory (Levico Terme, 1997), Lecture Notes in Pure and Appl. Math., 198, Dekker, New York, 1998, 303-314.  Google Scholar

[27]

(Russian) Izv. Ross. Akad. Nauk Ser. Mat., 66 (2002), 23-48; translation in Izv. Math., 66 (2002), 63-487. doi: 10.1070/IM2002v066n03ABEH000386.  Google Scholar

[28]

(Russian) Izv. Ross. Akad. Nauk Ser. Mat., 71 (2007), 3-18; translation in Izv. Math., 71 (2007), 657-672. doi: 10.1070/IM2007v071n04ABEH002371.  Google Scholar

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