Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Codimensions of polynomial identities of representations of Lie algebras


Author: A. S. Gordienko
Journal: Proc. Amer. Math. Soc. 141 (2013), 3369-3382
MSC (2010): Primary 17B01; Secondary 16R10, 17B10, 20C30
DOI: https://doi.org/10.1090/S0002-9939-2013-11622-9
Published electronically: June 18, 2013
MathSciNet review: 3080160
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider a representation $ \rho \colon L \to \mathfrak{gl}(V)$ where $ L$ is a Lie algebra and $ V$ is a finite dimensional vector space. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities of $ \rho $.


References [Enhancements On Off] (What's this?)

  • 1. Giambruno, A., Zaicev, M.V. (2005) Polynomial identities and asymptotic methods. AMS Mathematical Surveys and Monographs Vol. 122, Providence, RI, 352 pp. MR 2176105 (2006g:16054)
  • 2. Zaitsev, M.V. (2002) Integrality of exponents of growth of identities of finite-dimensional Lie algebras. Izv. Math. 66, 463-487. MR 1921808 (2003g:17004)
  • 3. Razmyslov, Yu.P. (1994) Identities of algebras and their representations, Transl. Math. Monogr., vol. 138, Amer. Math. Soc., Providence, RI. MR 1291603 (95i:16022)
  • 4. Plotkin, B.I., Vovsi, S.M. (1983) Varieties of group representations: general theory, connections, and applications. ``Zinante'', Riga (Russian). MR 739330 (86e:20001)
  • 5. Goto, M., Grosshans, F. (1978) Semisimple Lie algebras. Mercel Dekker, New York and Basel. MR 0573070 (58:28084)
  • 6. Bakhturin, Yu.A. (1987) Identical relations in Lie algebras. VNU Science Press, Utrecht. MR 886063 (88f:17032)
  • 7. Drensky, V.S. (2000) Free algebras and PI-algebras: graduate course in algebra. Springer-Verlag, Singapore. MR 1712064 (2000j:16002)
  • 8. Jacobson, N. (1989) Basic algebra II. Second edition. W.H. Freeman and Company, New York. MR 1009787 (90m:00007)
  • 9. Humphreys, J.E. (1978) Introduction to Lie algebras and representation theory. Springer-Verlag, New York. MR 499562 (81b:17007)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 17B01, 16R10, 17B10, 20C30

Retrieve articles in all journals with MSC (2010): 17B01, 16R10, 17B10, 20C30


Additional Information

A. S. Gordienko
Affiliation: Memorial University of Newfoundland, St. John’s, NL, Canada
Email: alexey.gordienko@vub.ac.be

DOI: https://doi.org/10.1090/S0002-9939-2013-11622-9
Keywords: Lie algebra, polynomial identity, codimension, cocharacter, symmetric group, Young diagram
Received by editor(s): June 17, 2011
Received by editor(s) in revised form: December 15, 2011
Published electronically: June 18, 2013
Additional Notes: This work was supported by postdoctoral fellowships from the Atlantic Association for Research in Mathematical Sciences (AARMS), the Atlantic Algebra Centre (AAC), Memorial University of Newfoundland (MUN), and the Natural Sciences and Engineering Research Council of Canada (NSERC)
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society