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
References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  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) Uspekhi Mat. Nauk, 45 (1990), 25-45, 189; translation in Russian Math. Surveys, 45 (1990), 27-52. 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, Comm. Algebra, 27 (1999), 2223-2230. doi: 10.1080/00927879908826560.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Pagon, D. Repovš and M. Zaicev, On the codimension growth of simple color Lie superalgebras, J. Lie Theory, 22 (2012), 465-479.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  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) Uspekhi Mat. Nauk, 45 (1990), 25-45, 189; translation in Russian Math. Surveys, 45 (1990), 27-52. 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, Comm. Algebra, 27 (1999), 2223-2230. doi: 10.1080/00927879908826560.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Pagon, D. Repovš and M. Zaicev, On the codimension growth of simple color Lie superalgebras, J. Lie Theory, 22 (2012), 465-479.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

A. V. Grishin. On non-Spechtianness of the variety of associative rings that satisfy the identity $x^{32} = 0$. Electronic Research Announcements, 2000, 6: 50-51.

[2]

Krzysztof Frączek. Polynomial growth of the derivative for diffeomorphisms on tori. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 489-516. doi: 10.3934/dcds.2004.11.489
[3]

Svetlana Pastukhova, Valeria Chiadò Piat. Homogenization of multivalued monotone operators with variable growth exponent. Networks & Heterogeneous Media, 2020, 15 (2) : 281-305. doi: 10.3934/nhm.2020013
[4]

A. Giambruno and M. Zaicev. Minimal varieties of algebras of exponential growth. Electronic Research Announcements, 2000, 6: 40-44.

[5]

Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197
[6]

Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921
[7]

Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131
[8]

Van Cyr, Bryna Kra. The automorphism group of a minimal shift of stretched exponential growth. Journal of Modern Dynamics, 2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483
[9]

Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1
[10]

Alaa Hayek, Serge Nicaise, Zaynab Salloum, Ali Wehbe. Exponential and polynomial stability results for networks of elastic and thermo-elastic rods. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021142
[11]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005
[12]

Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285
[13]

Joseph Bayara, André Conseibo, Artibano Micali, Moussa Ouattara. Derivations in power-associative algebras. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1359-1370. doi: 10.3934/dcdss.2011.4.1359
[14]

Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249
[15]

Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190
[16]

Yavdat Il'yasov. On critical exponent for an elliptic equation with non-Lipschitz nonlinearity. Conference Publications, 2011, 2011 (Special) : 698-706. doi: 10.3934/proc.2011.2011.698
[17]

Shengfan Zhou, Linshan Wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 399-412. doi: 10.3934/dcds.2003.9.399
[18]

Nguyen Lam, Guozhen Lu. Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2187-2205. doi: 10.3934/dcds.2012.32.2187
[19]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[20]

Federica Sani. A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 405-428. doi: 10.3934/cpaa.2013.12.405

2020 Impact Factor: 0.929

Tools

Metrics

  • PDF downloads (84)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy