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June  2020, 28(2): 853-859. doi: 10.3934/era.2020044

On existence of PI-exponents of unital algebras

1. 

Faculty of Mathematics and Physics, University of Ljubljana, & Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia

2. 

Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, 119992 Moscow, Russia

Received  April 2020 Published  June 2020

Fund Project: The first author was supported by the Slovenian Research Agency grants P1-0292, J1-8131, N1-0114, N1-0083, and N1-0064. The second author was supported by the Russian Science Foundation grant 16-11-10013Π

We construct a family of unital non-associative algebras $ \{T_\alpha\vert\; 2<\alpha\in\mathbb R\} $ such that $ \underline{exp}(T_\alpha) = 2 $, whereas $ \alpha\le\overline{exp}(T_\alpha)\le\alpha+1 $. In particular, it follows that ordinary PI-exponent of codimension growth of algebra $ T_\alpha $ does not exist for any $ \alpha> 2 $. This is the first example of a unital algebra whose PI-exponent does not exist.

Citation: Dušan D. Repovš, Mikhail V. Zaicev. On existence of PI-exponents of unital algebras. Electronic Research Archive, 2020, 28 (2) : 853-859. doi: 10.3934/era.2020044
References:
[1]

Y. A. Bahturin, Identical Relations in Lie Algebras, VNU Science Press, b.v., Utrecht, 1987.  Google Scholar

[2]

Y. 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]

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

[4]

A. GiambrunoI. 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

[5]

A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math., 140 (1998), 145-155.  doi: 10.1006/aima.1998.1766.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

S. P. Mishchenko, Growth of varieties of Lie algebras, Russian Math. Surveys, 45 (1990), 27-52.  doi: 10.1070/RM1990v045n06ABEH002710.  Google Scholar

[10]

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

[11]

D. Repovš and M. Zaicev, Numerical invariants of identities of unital algebras, Comm. Algebra, 43 (2015), 3823-3839.  doi: 10.1080/00927872.2014.924130.  Google Scholar

[12]

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

[13]

M. V. Zaicev, Growth of codimensions of metabelian algebras, Moscow Univ. Math. Bull., 72 (2017), 233-237.  doi: 10.3103/S0027132217060031.  Google Scholar

[14]

M. Zaicev, On existence of PI-exponents of codimension growth, Electron. Res. Announc. Math. Sci., 21 (2014), 113-119.  doi: 10.3934/era.2014.21.113.  Google Scholar

[15]

M. V. Zaĭtsev, Integrality of exponents of growth of identities of finite dimensional Lie algebras, Izv. Math., 66 (2002), 463-487.  doi: 10.1070/IM2002v066n03ABEH000386.  Google Scholar

[16]

M. V. Zaĭtsev, Identities of finite-dimensional unitary algebras, Algebra Logic, 50 (2011), 381-404.  doi: 10.1007/s10469-011-9151-8.  Google Scholar

[17]

M. V. Zaĭtsev and D. Repovsh, Exponential codimension growth of identities of unitary algebras, Sb. Math., 206 (2015), 1440-1462.  doi: 10.4213/sm8454.  Google Scholar

show all references

References:
[1]

Y. A. Bahturin, Identical Relations in Lie Algebras, VNU Science Press, b.v., Utrecht, 1987.  Google Scholar

[2]

Y. 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]

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

[4]

A. GiambrunoI. 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

[5]

A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math., 140 (1998), 145-155.  doi: 10.1006/aima.1998.1766.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

S. P. Mishchenko, Growth of varieties of Lie algebras, Russian Math. Surveys, 45 (1990), 27-52.  doi: 10.1070/RM1990v045n06ABEH002710.  Google Scholar

[10]

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

[11]

D. Repovš and M. Zaicev, Numerical invariants of identities of unital algebras, Comm. Algebra, 43 (2015), 3823-3839.  doi: 10.1080/00927872.2014.924130.  Google Scholar

[12]

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

[13]

M. V. Zaicev, Growth of codimensions of metabelian algebras, Moscow Univ. Math. Bull., 72 (2017), 233-237.  doi: 10.3103/S0027132217060031.  Google Scholar

[14]

M. Zaicev, On existence of PI-exponents of codimension growth, Electron. Res. Announc. Math. Sci., 21 (2014), 113-119.  doi: 10.3934/era.2014.21.113.  Google Scholar

[15]

M. V. Zaĭtsev, Integrality of exponents of growth of identities of finite dimensional Lie algebras, Izv. Math., 66 (2002), 463-487.  doi: 10.1070/IM2002v066n03ABEH000386.  Google Scholar

[16]

M. V. Zaĭtsev, Identities of finite-dimensional unitary algebras, Algebra Logic, 50 (2011), 381-404.  doi: 10.1007/s10469-011-9151-8.  Google Scholar

[17]

M. V. Zaĭtsev and D. Repovsh, Exponential codimension growth of identities of unitary algebras, Sb. Math., 206 (2015), 1440-1462.  doi: 10.4213/sm8454.  Google Scholar

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