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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn039, 10 pages, doi:10.1093/imrn/rnn039 published on April 28, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

An Upper Bound for the Lower Central Series Quotients of a Free Associative Algebra

Galyna Dobrovolska

Pavel Etingof

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Correspondence: Correspondence to be sent to: galyna{at}mit.edu

Feigin and Shoikhet conjectured in [4], Feigin and Shoikhet, Mathematical Research Letters, 14,2007,781–95 that successive quotients Bm(An) of the lower central series filtration of a free associative algebra An have polynomial growth. In this paper we give a proof of this conjecture, using the structure of a representation of Wn, the Lie algebra of polynomial vector fields on Formula , on Bm(An) which was defined in [4]. Moreover, we show that the number of squares in a Young diagram D corresponding to an irreducible Wn-module in the Jordan–Hölder series of Bm(An) is bounded above by the integer Formula , which allows us to confirm the structure of B3(A3) conjectured in [4].



References

  1. Dobrovolska G., Kim J., Ma X. On the lower central series of an associative algebra. (2007) preprint arXiv:0709.1905.
  2. Eisenbud D. Commutative Algebra with a View Toward Algebraic Geometry (1995) Berlin: Springer. Graduate Texts in Mathematics, 150.
  3. Feigin B., Fuks D. B. Cohomologies of Lie groups and Lie algebras. In: Lie Groups and Lie Algebras 2—Onishchik A. L., Vinberg E. B., eds. (2000) Berlin: Springer.
  4. Feigin B., Shoikhet B. On [A,A]/[A, [A,A]] and on a Wn-action on the consecutive commutators of free associative algebras. Mathematical Research Letters (2007) 14(5):781–95.[ISI]
  5. Fuks D. B. Cohomology of Infinite-Dimensional Lie Algebras (1986) New York: Consultants Bureau.
  6. Fulton W. Young Tableaux (1997) London Mathematical Society Student Texts, 35. Cambridge: Cambridge University Press.

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This Article
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Right arrow Articles by Etingof, P.
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