American Institute of Mathematical Sciences

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  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
References:
[1]

Yu. A. Bahturin, Identical Relations in Lie Algebras,, Translated from the Russian by Bahturin, (1987).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Berele, An example concerning the constant in the asymptotics of codimension sequences,, \emph{Comm. Algebra}, 38 (2010), 3506.  doi: 10.1080/00927870902939426.  Google Scholar

[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.  Google Scholar

[7]

V. Drensky, Free Algebras and PI-Algebras,, Graduate Course in Algebra, (2000).   Google Scholar

[8]

A. S. Dzhumadil'daev, Codimension growth and non-Koszulity of Novikov operad,, \emph{Comm. Algebra}, 39 (2011), 2943.  doi: 10.1080/00927870903386494.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods,, Mathematical Surveys and Monographs, (2005).  doi: 10.1090/surv/122.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. R. Kemer, The Spechtian nature of T-ideals whose condimensions have power growth,, (Russian) \emph{Sibirsk. Mat. Ž.}, 19 (1978), 54.   Google Scholar

[15]

D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra,, \emph{Trans. Amer. Math. Soc.}, 181 (1973), 429.   Google Scholar

[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.   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]

S. P. Mishchenko, Growth of varieties of Lie algebras,, (Russian) \emph{Uspekhi Mat. Nauk}, 45 (1990), 25.  doi: 10.1070/RM1990v045n06ABEH002710.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Pagon, D. Repovš and M. Zaicev, On the codimension growth of simple color Lie superalgebras,, \emph{J. Lie Theory}, 22 (2012), 465.   Google Scholar

[23]

A. Regev, Existence of identities in $A\otimes B$,, \emph{Israel J. Math.}, 11 (1972), 131.  doi: 10.1007/BF02762615.  Google Scholar

[24]

A. Regev, Codimensions and trace codimensions of matrices are asymptotically equal,, \emph{Israel J. Math.}, 47 (1984), 246.  doi: 10.1007/BF02760520.  Google Scholar

[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.   Google Scholar

[26]

M. V. Zaicev, Varieties and identities of affine Kac-Moody algebras,, in \emph{Methods in Ring Theory} (Levico Terme, (1997), 303.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

Yu. A. Bahturin, Identical Relations in Lie Algebras,, Translated from the Russian by Bahturin, (1987).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Berele, An example concerning the constant in the asymptotics of codimension sequences,, \emph{Comm. Algebra}, 38 (2010), 3506.  doi: 10.1080/00927870902939426.  Google Scholar

[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.  Google Scholar

[7]

V. Drensky, Free Algebras and PI-Algebras,, Graduate Course in Algebra, (2000).   Google Scholar

[8]

A. S. Dzhumadil'daev, Codimension growth and non-Koszulity of Novikov operad,, \emph{Comm. Algebra}, 39 (2011), 2943.  doi: 10.1080/00927870903386494.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods,, Mathematical Surveys and Monographs, (2005).  doi: 10.1090/surv/122.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. R. Kemer, The Spechtian nature of T-ideals whose condimensions have power growth,, (Russian) \emph{Sibirsk. Mat. Ž.}, 19 (1978), 54.   Google Scholar

[15]

D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra,, \emph{Trans. Amer. Math. Soc.}, 181 (1973), 429.   Google Scholar

[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.   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]

S. P. Mishchenko, Growth of varieties of Lie algebras,, (Russian) \emph{Uspekhi Mat. Nauk}, 45 (1990), 25.  doi: 10.1070/RM1990v045n06ABEH002710.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Pagon, D. Repovš and M. Zaicev, On the codimension growth of simple color Lie superalgebras,, \emph{J. Lie Theory}, 22 (2012), 465.   Google Scholar

[23]

A. Regev, Existence of identities in $A\otimes B$,, \emph{Israel J. Math.}, 11 (1972), 131.  doi: 10.1007/BF02762615.  Google Scholar

[24]

A. Regev, Codimensions and trace codimensions of matrices are asymptotically equal,, \emph{Israel J. Math.}, 47 (1984), 246.  doi: 10.1007/BF02760520.  Google Scholar

[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.   Google Scholar

[26]

M. V. Zaicev, Varieties and identities of affine Kac-Moody algebras,, in \emph{Methods in Ring Theory} (Levico Terme, (1997), 303.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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