# American Institute of Mathematical Sciences

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

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.
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) \emph{Vestnik Moskov. Univ. Ser. I Mat. Mekh.}, 1982 (): 63. [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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##### 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) \emph{Vestnik Moskov. Univ. Ser. I Mat. Mekh.}, 1982 (): 63. [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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