On existence of PI-exponents of codimension growth

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

On existence of PI-exponents of codimension growth

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

Abstract
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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

References:

[1]	Yu. A. Bahturin, Identical Relations in Lie Algebras,, Translated from the Russian by Bahturin, (1987).
[2]	Yu. Bahturin and V. Drensky, Graded polynomial identities of matrices,, \emph{Linear Algebra Appl.}, 357 (2002), 15. doi: 10.1016/S0024-3795(02)00356-7.
[3]	F. Benanti and I. Sviridova, Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras,, \emph{Israel J. Math.}, 156 (2006), 73. doi: 10.1007/BF02773825.
[4]	A. Berele, Properties of hook Schur functions with applications to p.i. algebras,, \emph{Adv. in Appl. Math.}, 41 (2008), 52. doi: 10.1016/j.aam.2007.03.002.
[5]	A. Berele, An example concerning the constant in the asymptotics of codimension sequences,, \emph{Comm. Algebra}, 38 (2010), 3506. doi: 10.1080/00927870902939426.
[6]	A. Berele and A. Regev, Codimensions of products and of intersections of verbally prime T-ideals,, \emph{Israel J. Math.}, 103 (1998), 17. doi: 10.1007/BF02762265.
[7]	V. Drensky, Free Algebras and PI-Algebras,, Graduate Course in Algebra, (2000).
[8]	A. S. Dzhumadil'daev, Codimension growth and non-Koszulity of Novikov operad,, \emph{Comm. Algebra}, 39 (2011), 2943. doi: 10.1080/00927870903386494.
[9]	A. Giambruno, I. Shestakov and M. Zaicev, Finite-dimensional non-associative algebras and codimension growth,, \emph{Adv. in Appl. Math.}, 47 (2011), 125. doi: 10.1016/j.aam.2010.04.007.
[10]	A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: An exact estimate,, \emph{Adv. Math.}, 142 (1999), 221. doi: 10.1006/aima.1998.1790.
[11]	A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods,, Mathematical Surveys and Monographs, (2005). doi: 10.1090/surv/122.
[12]	A. Giambruno and M. Zaicev, Codimension growth of special simple Jordan algebras,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 3107. doi: 10.1090/S0002-9947-09-04865-X.
[13]	A. Giambruno and M. Zaicev, On codimension growth of finite-dimensional Lie superalgebras,, \emph{J. Lond. Math. Soc. (2)}, 85 (2012), 534. doi: 10.1112/jlms/jdr059.
[14]	A. R. Kemer, The Spechtian nature of T-ideals whose condimensions have power growth,, (Russian) \emph{Sibirsk. Mat. Ž.}, 19 (1978), 54.
[15]	D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra,, \emph{Trans. Amer. Math. Soc.}, 181 (1973), 429.
[16]	V. N. Latyšev, On Regev's theorem on identities in a tensor product of PI-algebras,, (Russian) \emph{Uspehi Mat. Nauk}, 27 (1972), 213.
[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.
[18]	S. P. Mishchenko, Growth of varieties of Lie algebras,, (Russian) \emph{Uspekhi Mat. Nauk}, 45 (1990), 25. doi: 10.1070/RM1990v045n06ABEH002710.
[19]	S. P. Mishchenko and V. M. Petrogradsky, Exponents of varieties of Lie algebras with a nilpotent commutator subalgebra,, \emph{Comm. Algebra}, 27 (1999), 2223. doi: 10.1080/00927879908826560.
[20]	S. P. Mishchenko, V. M. Petrogradsky and A. Regev, Poisson PI algebras,, \emph{Trans. Amer. Math. Soc.}, 359 (2007), 4669. doi: 10.1090/S0002-9947-07-04008-1.
[21]	S. Mishchenko and A. Valenti, A Leibniz variety with almost polynomial growth,, \emph{J. Pure Appl. Algebra}, 202 (2005), 82. doi: 10.1016/j.jpaa.2005.01.013.
[22]	D. Pagon, D. Repovš and M. Zaicev, On the codimension growth of simple color Lie superalgebras,, \emph{J. Lie Theory}, 22 (2012), 465.
[23]	A. Regev, Existence of identities in $A\otimes B$,, \emph{Israel J. Math.}, 11 (1972), 131. doi: 10.1007/BF02762615.
[24]	A. Regev, Codimensions and trace codimensions of matrices are asymptotically equal,, \emph{Israel J. Math.}, 47 (1984), 246. doi: 10.1007/BF02760520.
[25]	I. B. Volichenko, Varieties of Lie algebras with identity $[[x_1,x_2,x_3],$ $ [x_4,x_5,x_6]]$ $= 0$ over a field of characteristic zero,, (Russian) \emph{Sibirsk. Mat. Zh.}, 25 (1984), 40.
[26]	M. V. Zaicev, Varieties and identities of affine Kac-Moody algebras,, in \emph{Methods in Ring Theory} (Levico Terme, (1997), 303.
[27]	M. V. Zaitsev, Integrality of exponents of growth of identities of finite-dimensional Lie algebras,, (Russian) \emph{Izv. Ross. Akad. Nauk Ser. Mat.}, 66 (2002), 23. doi: 10.1070/IM2002v066n03ABEH000386.
[28]	M. V. Zaitsev and S. P. Mishchenko, The growth of some varieties of Lie superalgebras,, (Russian) \emph{Izv. Ross. Akad. Nauk Ser. Mat.}, 71 (2007), 3. doi: 10.1070/IM2007v071n04ABEH002371.

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References:

[1]	Yu. A. Bahturin, Identical Relations in Lie Algebras,, Translated from the Russian by Bahturin, (1987).
[2]	Yu. Bahturin and V. Drensky, Graded polynomial identities of matrices,, \emph{Linear Algebra Appl.}, 357 (2002), 15. doi: 10.1016/S0024-3795(02)00356-7.
[3]	F. Benanti and I. Sviridova, Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras,, \emph{Israel J. Math.}, 156 (2006), 73. doi: 10.1007/BF02773825.
[4]	A. Berele, Properties of hook Schur functions with applications to p.i. algebras,, \emph{Adv. in Appl. Math.}, 41 (2008), 52. doi: 10.1016/j.aam.2007.03.002.
[5]	A. Berele, An example concerning the constant in the asymptotics of codimension sequences,, \emph{Comm. Algebra}, 38 (2010), 3506. doi: 10.1080/00927870902939426.
[6]	A. Berele and A. Regev, Codimensions of products and of intersections of verbally prime T-ideals,, \emph{Israel J. Math.}, 103 (1998), 17. doi: 10.1007/BF02762265.
[7]	V. Drensky, Free Algebras and PI-Algebras,, Graduate Course in Algebra, (2000).
[8]	A. S. Dzhumadil'daev, Codimension growth and non-Koszulity of Novikov operad,, \emph{Comm. Algebra}, 39 (2011), 2943. doi: 10.1080/00927870903386494.
[9]	A. Giambruno, I. Shestakov and M. Zaicev, Finite-dimensional non-associative algebras and codimension growth,, \emph{Adv. in Appl. Math.}, 47 (2011), 125. doi: 10.1016/j.aam.2010.04.007.
[10]	A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: An exact estimate,, \emph{Adv. Math.}, 142 (1999), 221. doi: 10.1006/aima.1998.1790.
[11]	A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods,, Mathematical Surveys and Monographs, (2005). doi: 10.1090/surv/122.
[12]	A. Giambruno and M. Zaicev, Codimension growth of special simple Jordan algebras,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 3107. doi: 10.1090/S0002-9947-09-04865-X.
[13]	A. Giambruno and M. Zaicev, On codimension growth of finite-dimensional Lie superalgebras,, \emph{J. Lond. Math. Soc. (2)}, 85 (2012), 534. doi: 10.1112/jlms/jdr059.
[14]	A. R. Kemer, The Spechtian nature of T-ideals whose condimensions have power growth,, (Russian) \emph{Sibirsk. Mat. Ž.}, 19 (1978), 54.
[15]	D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra,, \emph{Trans. Amer. Math. Soc.}, 181 (1973), 429.
[16]	V. N. Latyšev, On Regev's theorem on identities in a tensor product of PI-algebras,, (Russian) \emph{Uspehi Mat. Nauk}, 27 (1972), 213.
[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.
[18]	S. P. Mishchenko, Growth of varieties of Lie algebras,, (Russian) \emph{Uspekhi Mat. Nauk}, 45 (1990), 25. doi: 10.1070/RM1990v045n06ABEH002710.
[19]	S. P. Mishchenko and V. M. Petrogradsky, Exponents of varieties of Lie algebras with a nilpotent commutator subalgebra,, \emph{Comm. Algebra}, 27 (1999), 2223. doi: 10.1080/00927879908826560.
[20]	S. P. Mishchenko, V. M. Petrogradsky and A. Regev, Poisson PI algebras,, \emph{Trans. Amer. Math. Soc.}, 359 (2007), 4669. doi: 10.1090/S0002-9947-07-04008-1.
[21]	S. Mishchenko and A. Valenti, A Leibniz variety with almost polynomial growth,, \emph{J. Pure Appl. Algebra}, 202 (2005), 82. doi: 10.1016/j.jpaa.2005.01.013.
[22]	D. Pagon, D. Repovš and M. Zaicev, On the codimension growth of simple color Lie superalgebras,, \emph{J. Lie Theory}, 22 (2012), 465.
[23]	A. Regev, Existence of identities in $A\otimes B$,, \emph{Israel J. Math.}, 11 (1972), 131. doi: 10.1007/BF02762615.
[24]	A. Regev, Codimensions and trace codimensions of matrices are asymptotically equal,, \emph{Israel J. Math.}, 47 (1984), 246. doi: 10.1007/BF02760520.
[25]	I. B. Volichenko, Varieties of Lie algebras with identity $[[x_1,x_2,x_3],$ $ [x_4,x_5,x_6]]$ $= 0$ over a field of characteristic zero,, (Russian) \emph{Sibirsk. Mat. Zh.}, 25 (1984), 40.
[26]	M. V. Zaicev, Varieties and identities of affine Kac-Moody algebras,, in \emph{Methods in Ring Theory} (Levico Terme, (1997), 303.
[27]	M. V. Zaitsev, Integrality of exponents of growth of identities of finite-dimensional Lie algebras,, (Russian) \emph{Izv. Ross. Akad. Nauk Ser. Mat.}, 66 (2002), 23. doi: 10.1070/IM2002v066n03ABEH000386.
[28]	M. V. Zaitsev and S. P. Mishchenko, The growth of some varieties of Lie superalgebras,, (Russian) \emph{Izv. Ross. Akad. Nauk Ser. Mat.}, 71 (2007), 3. doi: 10.1070/IM2007v071n04ABEH002371.

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