RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Izv. RAN. Ser. Mat.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 3, Pages 23–48 (Mi izv386)  

This article is cited in 51 scientific papers (total in 51 papers)

Integrality of exponents of codimension growth of finite-dimensional Lie algebras

M. V. Zaicev

M. V. Lomonosov Moscow State University

Abstract: We study the asymptotic behaviour of the codimension growth sequence $c_n(L)$ of a finite-dimensional Lie algebra $L$ over a field of characteristic zero. It is known that the growth of the sequence $\{c_n(L)\}$ is bounded by an exponential function of $n$, and hence there exist the upper and lower limits of the $n$th roots of $c_n(L)$, which are called the upper and lower exponents. By Amitsur's conjecture, the upper and lower exponents should coincide and be integers. This conjecture has been confirmed in the associative case for any PI-algebra. For finite-dimensional Lie algebras, a positive solution has been found for soluble, simple and semisimple algebras and also for algebras whose soluble radical is nilpotent. For infinite-dimensional Lie algebras, the problem has been solved in the negative. In this paper we give a proof of Amitsur's conjecture for arbitrary finite-dimensional Lie algebras.

DOI: https://doi.org/10.4213/im386

Full text: PDF file (2259 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2002, 66:3, 463–487

Bibliographic databases:

UDC: 512.8
MSC: Primary 17B01, 17B20, 16R10; Secondary 20C30, 17C05
Received: 28.02.2001

Citation: M. V. Zaicev, “Integrality of exponents of codimension growth of finite-dimensional Lie algebras”, Izv. RAN. Ser. Mat., 66:3 (2002), 23–48; Izv. Math., 66:3 (2002), 463–487

Citation in format AMSBIB
\Bibitem{Zai02}
\by M.~V.~Zaicev
\paper Integrality of exponents of codimension growth of finite-dimensional Lie algebras
\jour Izv. RAN. Ser. Mat.
\yr 2002
\vol 66
\issue 3
\pages 23--48
\mathnet{http://mi.mathnet.ru/izv386}
\crossref{https://doi.org/10.4213/im386}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1921808}
\zmath{https://zbmath.org/?q=an:1057.17003}
\elib{http://elibrary.ru/item.asp?id=14121485}
\transl
\jour Izv. Math.
\yr 2002
\vol 66
\issue 3
\pages 463--487
\crossref{https://doi.org/10.1070/IM2002v066n03ABEH000386}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33748517655}


Linking options:
  • http://mi.mathnet.ru/eng/izv386
  • https://doi.org/10.4213/im386
  • http://mi.mathnet.ru/eng/izv/v66/i3/p23

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Giambruno A., Mishchenko S., Zaicev M., “Algebras with Intermediate Growth of the Codimensions”, Adv. Appl. Math., 37:3 (2006), 360–377  crossref  mathscinet  zmath  isi  elib  scopus
    2. Giambruno A., Mishchenko S., Zaicev M., “Codimension growth of two-dimensional non-associative algebras”, Proc Amer Math Soc, 135:11 (2007), 3405–3415  crossref  mathscinet  zmath  isi  elib  scopus
    3. Giambruno A., Zaicev M., “Multialternating Jordan polynomials and codimension growth of matrix algebras”, Linear Algebra Appl, 422:2–3 (2007), 372–379  crossref  mathscinet  zmath  isi  elib  scopus
    4. M. V. Zaitsev, S. P. Mishchenko, “Identities for Lie superalgebras with a nilpotent commutator subalgebra”, Algebra and Logic, 47:5 (2008), 348–364  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    5. S. S. Mishchenko, “On growth of varieties of commutative linear algebras”, J. Math. Sci., 163:6 (2009), 739–742  mathnet  crossref  mathscinet  elib  elib
    6. Giambruno, A, “Proper identities, Lie identities and exponential codimension growth”, Journal of Algebra, 320:5 (2008), 1933  crossref  mathscinet  zmath  isi  elib  scopus
    7. Giambruno A., Mishchenko S., Zaicev M., “Codimensions of algebras and growth functions”, Advances in Mathematics, 217:3 (2008), 1027–1052  crossref  mathscinet  zmath  isi  elib  scopus
    8. Mishchenko S., Valenti A., “On the growth of varieties of algebras”, Groups, Rings and Group Rings, Contemporary Mathematics, 499, 2009, 229–243  crossref  mathscinet  zmath  isi
    9. Giambruno A., Mishchenko S., Zaicev M., “Polynomial Identities of Algebras of Small Dimension”, Comm Algebra, 37:6 (2009), 1934–1948  crossref  mathscinet  zmath  isi  elib  scopus
    10. Mishchenko S., Valenti A., “Varieties with at most quadratic growth”, Israel Journal of Mathematics, 178:1 (2010), 209–228  crossref  mathscinet  zmath  isi  scopus
    11. Aljadeff E., “On the Codimension Growth of G-Graded Algebras”, Proceedings of the American Mathematical Society, 138:7 (2010), 2311–2320  crossref  mathscinet  zmath  isi  scopus
    12. Giambruno A., Zaicev M., “Codimension Growth of Special Simple Jordan Algebras”, Trans Amer Math Soc, 362:6 (2010), 3107–3123  crossref  mathscinet  zmath  isi  elib  scopus
    13. O. E. Bezushchak, M. V. Zaitsev, “Exponents of Identities of Group Rings”, Math. Notes, 89:5 (2011), 605–612  mathnet  crossref  crossref  mathscinet  isi
    14. M. V. Zaitsev, “Identities of unitary finite-dimensional algebras”, Algebra and Logic, 50:5 (2011), 381–404  mathnet  crossref  mathscinet  zmath  isi
    15. Giambruno A., Shestakov I., Zaicev M., “Finite-dimensional non-associative algebras and codimension growth”, Adv in Appl Math, 47:1 (2011), 125–139  crossref  mathscinet  zmath  isi  scopus
    16. Aljadeff E., Giambruno A., La Mattina D., “Graded polynomial identities and exponential growth”, J Reine Angew Math, 650 (2011), 83–100  crossref  mathscinet  zmath  isi  elib  scopus
    17. Verevkin A.B., Zaitsev M.V., Mischenko S.P., “Dostatochnoe uslovie sovpadeniya nizhnei i verkhnei eksponent mnogoobraziya lineinykh algebr”, Vestnik Moskovskogo universiteta. Seriya 1: Matematika. Mekhanika, 2011, no. 2, 36–39  mathscinet  zmath  elib
    18. Mischenko S.S., “Novyi primer mnogoobraziya algebr li s drobnoi eksponentoi”, Vestnik moskovskogo universiteta. seriya 1: matematika. mekhanika, 2011, no. 6, 44–47  elib
    19. A.S. Gordienko, “Graded polynomial identities, group actions, and exponential growth of Lie algebras”, Journal of Algebra, 367 (2012), 26  crossref  mathscinet  zmath  isi  scopus
    20. Aljadeff E., “Group Graded Pi-Algebras and their Codimension Growth”, Isr. J. Math., 189:1 (2012), 189–205  crossref  mathscinet  zmath  isi  elib  scopus
    21. Di Vincenzo O.M. Nardozza V., “On the Existence of the Graded Exponent for Finite Dimensional Z(P)-Graded Algebras”, Can. Math. Bul.-Bul. Can. Math., 55:2 (2012), 271–284  crossref  mathscinet  zmath  isi  scopus
    22. Giambruno A., Zaicev M., “On Codimension Growth of Finite-Dimensional Lie Superalgebras”, J. Lond. Math. Soc.-Second Ser., 85:Part 2 (2012), 534–548  crossref  mathscinet  zmath  isi  scopus
    23. Pagon D., Repovs D., Zaicev M., “On the Codimension Growth of Simple Color Lie Superalgebras”, J. Lie Theory, 22:2 (2012), 465–479  mathscinet  zmath  isi  elib
    24. A.S. Gordienko, “Amitsur’s conjecture for associative algebras with a generalized Hopf action”, Journal of Pure and Applied Algebra, 2013  crossref  mathscinet  isi  scopus
    25. A. S. Gordienko, M. V. Kochetov, “Derivations, Gradings, Actions of Algebraic Groups, and Codimension Growth of Polynomial Identities”, Algebr Represent Theor, 2013  crossref  mathscinet  isi  scopus
    26. A. S. GORDIENKO, “ON A FORMULA FOR THE PI-EXPONENT OF Lie ALGEBRAS”, J. Algebra Appl, 2013, 1350069  crossref  mathscinet  isi  scopus
    27. Antonio Giambruno, M.d.a.S.ilva Souza, “Minimal varieties of graded Lie algebras of exponential growth and the special Lie algebra”, Journal of Pure and Applied Algebra, 2013  crossref  mathscinet  isi  scopus
    28. Repovs D. Zaicev M., “On Identities of Infinite Dimensional Lie Superalgebras”, Proc. Amer. Math. Soc., 141:12 (2013), 4139–4153  crossref  mathscinet  zmath  isi  elib  scopus
    29. Giambruno A., Zaicev M., “Non-Integrality of the Pi-Exponent of Special Lie Algebras”, Adv. Appl. Math., 51:5 (2013), 619–634  crossref  mathscinet  zmath  isi  scopus
    30. Gordienko A.S., “Codimensions of Polynomial Identities of Representations of Lie Algebras”, Proc. Amer. Math. Soc., 141:10 (2013), 3369–3382  crossref  mathscinet  zmath  isi  scopus
    31. Repovs D., Zaicev M., “On the Codimension Growth of Almost Nilpotent Lie Algebras”, Isr. J. Math., 194:1 (2013), 137–150  crossref  mathscinet  zmath  isi  elib  scopus
    32. Gordienko A.S., “The Structure of H-(Co)Module Lie Algebras”, J. Lie Theory, 23:3 (2013), 669–689  mathscinet  zmath  isi  elib
    33. S. P. Mishchenko, O. A. Bogdanchuk, “PI-exponents of some simple algebras with unit”, J. Math. Sci., 206:6 (2015), 688–693  mathnet  crossref  mathscinet
    34. O. E. Bezushchak, A. A. Beljaev, M. V. Zaicev, “Codimension growth of algebras with adjoint unit”, J. Math. Sci., 206:5 (2015), 462–473  mathnet  crossref  mathscinet
    35. M. V. Zaicev, D. Repovš, “A Four-Dimensional Simple Algebra with Fractional PI-Exponent”, Math. Notes, 95:4 (2014), 487–499  mathnet  crossref  crossref  mathscinet  isi  elib
    36. Mikhail Zaicev, “On existence of PI-exponents of codimension growth”, ERA-MS, 21 (2014), 113  crossref  mathscinet  zmath
    37. Belov-Kanel A. Giambruno A. Rowen L.H. Vishne U., “Zariski Closed Algebras in Varieties of Universal Algebra”, Algebr. Represent. Theory, 17:6 (2014), 1771–1783  crossref  mathscinet  zmath  isi  scopus
    38. Repovs D. Zaicev M., “Graded Identities of Some Simple Lie Superalgebras”, Algebr. Represent. Theory, 17:5 (2014), 1401–1412  crossref  mathscinet  zmath  isi  scopus
    39. Repovs D., Zaicev M., “Graded Codimensions of Lie Superalgebra B(2)”, J. Algebra, 422 (2015), 1–10  crossref  mathscinet  zmath  isi  scopus
    40. Gordienko A.S., “Amitsur's Conjecture For Polynomial H-Identities of H-Module Lie Algebras”, Trans. Am. Math. Soc., 367:1 (2015), 313–354  crossref  mathscinet  zmath  isi
    41. Repovs D., Zaicev M., “Numerical Invariants of Identities of Unital Algebras”, Commun. Algebr., 43:9 (2015), 3823–3839  crossref  mathscinet  zmath  isi  elib  scopus
    42. Repovs D., Zaicev M., “Graded PI-exponents of simple Lie superalgebras”, Ark. Mat., 54:1 (2016), 147–156  crossref  mathscinet  zmath  isi  elib  scopus
    43. Repovs D.D., Zaicev M.V., “Pauli gradings on Lie superalgebras and graded codimension growth”, Linear Alg. Appl., 520 (2017), 134–150  crossref  mathscinet  zmath  isi  scopus
    44. Repovs D., Zaicev M., “Identities of graded simple algebras”, Linear Multilinear Algebra, 65:1 (2017), 44–57  crossref  mathscinet  zmath  isi  scopus
    45. Gordienko A.S., “Actions of Ore Extensions and Growth of Polynomial H-Identities”, Commun. Algebr., 46:7 (2018), 3014–3032  crossref  mathscinet  zmath  isi  scopus
    46. Repovs D.D., Zaicev M.V., “Z(2)-Graded Codimensions of Unital Algebras”, Int. J. Algebr. Comput., 28:3 (2018), 483–500  crossref  mathscinet  zmath  isi  scopus
    47. Repovs D.D., Zaicev V M., “Codimension Growth of Solvable Lie Superalgebras”, J. Lie Theory, 28:4 (2018), 1189–1199  mathscinet  zmath  isi
    48. Gordienko A., “Lie Algebras Simple With Respect to a Taft Algebra Action”, J. Algebra, 517 (2019), 249–275  crossref  mathscinet  zmath  isi  scopus
    49. Mishchenko S., Valenti A., “Varieties With At Most Cubic Growth”, J. Algebra, 518 (2019), 321–342  crossref  mathscinet  zmath  isi  scopus
    50. M. V. Zaitsev, D. D. Repovsh, “Kombinatorika dvoichnykh slov i koraz­mernosti tozhdestv levonilpotentnykh algebr”, Algebra i logika, 58:1 (2019), 35–51  mathnet  crossref
    51. M. V. Zaitsev, S. P. Mischenko, “Posledovatelnosti korazmernostei tozhdestv i ikh asimptoticheskoe povedenie”, Fundament. i prikl. matem., 22:4 (2019), 115–127  mathnet
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
    Number of views:
    This page:368
    Full text:120
    References:48
    First page:3

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020