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, VNU Science Press, b.v., Utrecht, 1987.

[2]

Yu. Bahturin and V. Drensky, Graded polynomial identities of matrices, Linear Algebra Appl., 357 (2002), 15-34. 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, Israel J. Math., 156 (2006), 73-91. doi: 10.1007/BF02773825.

[4]

A. Berele, Properties of hook Schur functions with applications to p.i. algebras, Adv. in Appl. Math., 41 (2008), 52-75. doi: 10.1016/j.aam.2007.03.002.

[5]

A. Berele, An example concerning the constant in the asymptotics of codimension sequences, Comm. Algebra, 38 (2010), 3506-3510. doi: 10.1080/00927870902939426.

[6]

A. Berele and A. Regev, Codimensions of products and of intersections of verbally prime T-ideals, Israel J. Math., 103 (1998), 17-28. doi: 10.1007/BF02762265.

[7]

V. Drensky, Free Algebras and PI-Algebras, Graduate Course in Algebra, Springer-Verlag Singapore, Singapore, 2000.

[8]

A. S. Dzhumadil'daev, Codimension growth and non-Koszulity of Novikov operad, Comm. Algebra, 39 (2011), 2943-2952. doi: 10.1080/00927870903386494.

[9]

A. Giambruno, I. Shestakov and M. Zaicev, Finite-dimensional non-associative algebras and codimension growth, Adv. in Appl. Math., 47 (2011), 125-139. doi: 10.1016/j.aam.2010.04.007.

[10]

A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: An exact estimate, Adv. Math., 142 (1999), 221-243. doi: 10.1006/aima.1998.1790.

[11]

A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Mathematical Surveys and Monographs, 122, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/122.

[12]

A. Giambruno and M. Zaicev, Codimension growth of special simple Jordan algebras, Trans. Amer. Math. Soc., 362 (2010), 3107-3123. doi: 10.1090/S0002-9947-09-04865-X.

[13]

A. Giambruno and M. Zaicev, On codimension growth of finite-dimensional Lie superalgebras, J. Lond. Math. Soc. (2), 85 (2012), 534-548. doi: 10.1112/jlms/jdr059.

[14]

A. R. Kemer, The Spechtian nature of T-ideals whose condimensions have power growth, (Russian) Sibirsk. Mat. Ž., 19 (1978), 54-69, 237.

[15]

D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc., 181 (1973), 429-438.

[16]

V. N. Latyšev, On Regev's theorem on identities in a tensor product of PI-algebras, (Russian) Uspehi Mat. Nauk, 27 (1972), 213-214.

[17]

S. P. Mishchenko, Varieties of Lie algebras with weak growth of the sequence of codimensions, (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1982, 63-66.

[18]

S. P. Mishchenko, Growth of varieties of Lie algebras, (Russian) Uspekhi Mat. Nauk, 45 (1990), 25-45, 189; translation in Russian Math. Surveys, 45 (1990), 27-52. doi: 10.1070/RM1990v045n06ABEH002710.

[19]

S. P. Mishchenko and V. M. Petrogradsky, Exponents of varieties of Lie algebras with a nilpotent commutator subalgebra, Comm. Algebra, 27 (1999), 2223-2230. doi: 10.1080/00927879908826560.

[20]

S. P. Mishchenko, V. M. Petrogradsky and A. Regev, Poisson PI algebras, Trans. Amer. Math. Soc., 359 (2007), 4669-4694. doi: 10.1090/S0002-9947-07-04008-1.

[21]

S. Mishchenko and A. Valenti, A Leibniz variety with almost polynomial growth, J. Pure Appl. Algebra, 202 (2005), 82-101. 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, J. Lie Theory, 22 (2012), 465-479.

[23]

A. Regev, Existence of identities in $A\otimes B$, Israel J. Math., 11 (1972), 131-152. doi: 10.1007/BF02762615.

[24]

A. Regev, Codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math., 47 (1984), 246-250. 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) Sibirsk. Mat. Zh., 25 (1984), 40-54.

[26]

M. V. Zaicev, Varieties and identities of affine Kac-Moody algebras, in Methods in Ring Theory (Levico Terme, 1997), Lecture Notes in Pure and Appl. Math., 198, Dekker, New York, 1998, 303-314.

[27]

M. V. Zaitsev, Integrality of exponents of growth of identities of finite-dimensional Lie algebras, (Russian) Izv. Ross. Akad. Nauk Ser. Mat., 66 (2002), 23-48; translation in Izv. Math., 66 (2002), 63-487. doi: 10.1070/IM2002v066n03ABEH000386.

[28]

M. V. Zaitsev and S. P. Mishchenko, The growth of some varieties of Lie superalgebras, (Russian) Izv. Ross. Akad. Nauk Ser. Mat., 71 (2007), 3-18; translation in Izv. Math., 71 (2007), 657-672. doi: 10.1070/IM2007v071n04ABEH002371.

show all references

References:
[1]

Yu. A. Bahturin, Identical Relations in Lie Algebras, Translated from the Russian by Bahturin, VNU Science Press, b.v., Utrecht, 1987.

[2]

Yu. Bahturin and V. Drensky, Graded polynomial identities of matrices, Linear Algebra Appl., 357 (2002), 15-34. 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, Israel J. Math., 156 (2006), 73-91. doi: 10.1007/BF02773825.

[4]

A. Berele, Properties of hook Schur functions with applications to p.i. algebras, Adv. in Appl. Math., 41 (2008), 52-75. doi: 10.1016/j.aam.2007.03.002.

[5]

A. Berele, An example concerning the constant in the asymptotics of codimension sequences, Comm. Algebra, 38 (2010), 3506-3510. doi: 10.1080/00927870902939426.

[6]

A. Berele and A. Regev, Codimensions of products and of intersections of verbally prime T-ideals, Israel J. Math., 103 (1998), 17-28. doi: 10.1007/BF02762265.

[7]

V. Drensky, Free Algebras and PI-Algebras, Graduate Course in Algebra, Springer-Verlag Singapore, Singapore, 2000.

[8]

A. S. Dzhumadil'daev, Codimension growth and non-Koszulity of Novikov operad, Comm. Algebra, 39 (2011), 2943-2952. doi: 10.1080/00927870903386494.

[9]

A. Giambruno, I. Shestakov and M. Zaicev, Finite-dimensional non-associative algebras and codimension growth, Adv. in Appl. Math., 47 (2011), 125-139. doi: 10.1016/j.aam.2010.04.007.

[10]

A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: An exact estimate, Adv. Math., 142 (1999), 221-243. doi: 10.1006/aima.1998.1790.

[11]

A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Mathematical Surveys and Monographs, 122, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/122.

[12]

A. Giambruno and M. Zaicev, Codimension growth of special simple Jordan algebras, Trans. Amer. Math. Soc., 362 (2010), 3107-3123. doi: 10.1090/S0002-9947-09-04865-X.

[13]

A. Giambruno and M. Zaicev, On codimension growth of finite-dimensional Lie superalgebras, J. Lond. Math. Soc. (2), 85 (2012), 534-548. doi: 10.1112/jlms/jdr059.

[14]

A. R. Kemer, The Spechtian nature of T-ideals whose condimensions have power growth, (Russian) Sibirsk. Mat. Ž., 19 (1978), 54-69, 237.

[15]

D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc., 181 (1973), 429-438.

[16]

V. N. Latyšev, On Regev's theorem on identities in a tensor product of PI-algebras, (Russian) Uspehi Mat. Nauk, 27 (1972), 213-214.

[17]

S. P. Mishchenko, Varieties of Lie algebras with weak growth of the sequence of codimensions, (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1982, 63-66.

[18]

S. P. Mishchenko, Growth of varieties of Lie algebras, (Russian) Uspekhi Mat. Nauk, 45 (1990), 25-45, 189; translation in Russian Math. Surveys, 45 (1990), 27-52. doi: 10.1070/RM1990v045n06ABEH002710.

[19]

S. P. Mishchenko and V. M. Petrogradsky, Exponents of varieties of Lie algebras with a nilpotent commutator subalgebra, Comm. Algebra, 27 (1999), 2223-2230. doi: 10.1080/00927879908826560.

[20]

S. P. Mishchenko, V. M. Petrogradsky and A. Regev, Poisson PI algebras, Trans. Amer. Math. Soc., 359 (2007), 4669-4694. doi: 10.1090/S0002-9947-07-04008-1.

[21]

S. Mishchenko and A. Valenti, A Leibniz variety with almost polynomial growth, J. Pure Appl. Algebra, 202 (2005), 82-101. 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, J. Lie Theory, 22 (2012), 465-479.

[23]

A. Regev, Existence of identities in $A\otimes B$, Israel J. Math., 11 (1972), 131-152. doi: 10.1007/BF02762615.

[24]

A. Regev, Codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math., 47 (1984), 246-250. 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) Sibirsk. Mat. Zh., 25 (1984), 40-54.

[26]

M. V. Zaicev, Varieties and identities of affine Kac-Moody algebras, in Methods in Ring Theory (Levico Terme, 1997), Lecture Notes in Pure and Appl. Math., 198, Dekker, New York, 1998, 303-314.

[27]

M. V. Zaitsev, Integrality of exponents of growth of identities of finite-dimensional Lie algebras, (Russian) Izv. Ross. Akad. Nauk Ser. Mat., 66 (2002), 23-48; translation in Izv. Math., 66 (2002), 63-487. doi: 10.1070/IM2002v066n03ABEH000386.

[28]

M. V. Zaitsev and S. P. Mishchenko, The growth of some varieties of Lie superalgebras, (Russian) Izv. Ross. Akad. Nauk Ser. Mat., 71 (2007), 3-18; translation in Izv. Math., 71 (2007), 657-672. doi: 10.1070/IM2007v071n04ABEH002371.

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