Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Gröbner-Shirshov bases of the Lie algebra $ B_n^+$


Author: A. N. Koryukin
Translated by: A. V. Yakovlev
Original publication: Algebra i Analiz, tom 20 (2008), nomer 1.
Journal: St. Petersburg Math. J. 20 (2009), 65-94
MSC (2000): Primary 17Bxx
DOI: https://doi.org/10.1090/S1061-0022-08-01038-8
Published electronically: November 13, 2008
MathSciNet review: 2411970
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The minimal Gröbner-Shirshov bases of the positive part $ B_n^+$ of a simple finite-dimensional Lie algebra $ B_n$ over an arbitrary field of characteristic 0 are calculated, for the generators associated with simple roots and for an arbitrary ordering of these generators (i.e., an arbitrary basis of the $ n!$ Gröbner-Shirshov bases is chosen and studied). This is a completely new class of problems; until now, this program was carried out only for the Lie algebra $ A_n^+$. The minimal Gröbner-Shirshov basis of the Lie algebra $ B_n^+$ was calculated earlier by Bokut and Klein, but this was done for only one ordering of generators.


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

  • 1. A. I. Shirshov, On free Lie rings, Mat. Sb. (N.S.) 45 (1958), no. 2, 113-122. (Russian) MR 0099356 (20:5796)
  • 2. R. Lyndon, On Burnside's problem, Trans. Amer. Math. Soc. 77 (1954), 202-215. MR 0064049 (16:218b)
  • 3. A. N. Koryukin, Gröbner-Shirshov bases for the Lie algebra $ A_n$, Algebra i Logika 44 (2005), no. 2, 131-147; English transl., Algebra Logic 44 (2005), no. 2, 73-81. MR 2170693 (2006g:17015)
  • 4. A. N. Koryukin and K. P. Shum, Reduced bases of the Lie algebras $ D_n^+$, Sib. Zh. Ind. Mat. 9 (2006), no. 4, 90-104. (Russian) MR 2304801 (2007k:17009)
  • 5. L. A. Bokut and A. A. Klein, Serre relations and Gröbner-Shirshov bases for simple Lie algebras. I, Internat. J. Algebra Comput. 6 (1996), no. 4, 389-400. MR 1414346 (97k:17005)
  • 6. P. Lalonde and A. Ram, Standard Lyndon bases of Lie algebras and enveloping algebras, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1821-1830. MR 1273505 (95h:17013)
  • 7. L. A. Bokut and A. A. Klein, Serre relations and Gröbner-Shirshov bases for simple Lie algebras. II, Internat. J. Algebra Comput. 6 (1996), no. 4, 401-412. MR 1414346 (97k:17005)
  • 8. J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math., vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842 (48:2197)
  • 9. N. Jacobson, Lie algebras, Intersci. Tracts in Pure Appl. Math., No. 10, Intersci. Publ., New York-London, 1962. MR 0143793 (26:1345)
  • 10. N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6, Springer-Verlag, Berlin, 2002. MR 1890629 (2003a:17001)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 17Bxx

Retrieve articles in all journals with MSC (2000): 17Bxx


Additional Information

A. N. Koryukin
Affiliation: Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, 4 Academician Koptyug Avenue, 630090, Novosibirsk, Russia
Email: koryukin@ngs.ru

DOI: https://doi.org/10.1090/S1061-0022-08-01038-8
Received by editor(s): January 29, 2007
Published electronically: November 13, 2008
Additional Notes: The work was partially supported by RFBR (grant no. 05-01-00230), by the Leading Scientific Schools Foundation (grant no. 2069.20031), and by the Complex Integration Projects Foundation of the Siberian Branch of RAS
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society