On existence of PI-exponents of unital algebras

Dušan D. Repovš; Mikhail V. Zaicev; Dušan D. Repovš; Mikhail V. Zaicev

doi:10.3934/era.2020044

Electronic Research Archive

2020, Volume 28, Issue 2: 853-859. doi: 10.3934/era.2020044

Previous Article Next Article

On existence of PI-exponents of unital algebras

Dušan D. Repovš ^{1
,},
Mikhail V. Zaicev ^{2
,}

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: 01 April 2020
Primary: 16R10; Secondary: 16P90

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.
- Polynomial identities,
- exponential codimension growth,
- PI-exponent,
- unital algebra,
- numerical invariant
Citation: Dušan D. Repovš, Mikhail V. Zaicev. On existence of PI-exponents of unital algebras[J]. Electronic Research Archive, 2020, 28(2): 853-859. doi: 10.3934/era.2020044

Related Papers:

Abstract

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.

References

[1]	Y. A. Bahturin, Identical Relations in Lie Algebras, VNU Science Press, b.v., Utrecht, 1987.
[2]	Graded polynomial identities of matrices. Linear Algebra Appl. (2002) 357: 15-34.
[3]	V. Drensky, Free Algebras and PI-Algebras, Graduate Course in Algebra, Springer-Verlag Singapore, Singapore, 2000.
[4]	Finite-dimensional non-associative algebras and codimension growth. Adv. in Appl. Math. (2011) 47: 125-139.
[5]	On codimension growth of finitely generated associative algebras. Adv. Math. (1998) 140: 145-155.
[6]	Exponential codimension growth of PI algebras: An exact estimate. Adv. Math. (1999) 142: 221-243.
[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
[8]	On codimension growth of finite-dimensional Lie superalgebras. J. Lond. Math. Soc. (2) (2012) 85: 534-548.
[9]	Growth of varieties of Lie algebras. Russian Math. Surveys (1990) 45: 27-52.
[10]	Existence of identities in $A \otimes B$. Israel J. Math. (1972) 11: 131-152.
[11]	Numerical invariants of identities of unital algebras. Comm. Algebra (2015) 43: 3823-3839.
[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.
[13]	Growth of codimensions of metabelian algebras. Moscow Univ. Math. Bull. (2017) 72: 233-237.
[14]	On existence of PI-exponents of codimension growth. Electron. Res. Announc. Math. Sci. (2014) 21: 113-119.
[15]	Integrality of exponents of growth of identities of finite dimensional Lie algebras. Izv. Math. (2002) 66: 463-487.
[16]	Identities of finite-dimensional unitary algebras. Algebra Logic (2011) 50: 381-404.
[17]	Exponential codimension growth of identities of unitary algebras. Sb. Math. (2015) 206: 1440-1462.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)