Skip to main content
Log in

Bases for partially commutative Lie algebras

  • Published:
Algebra and Logic Aims and scope

We give an explicit description of linear bases for partially commutative Lie algebras. To do this, use is made of the Gröbner–Shirshov basis method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G. Duchamp and D. Krob, “Free partially commutative structures,” J. Alg., 156, No. 2, 318-361 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Servatius, “Automorphisms of graph groups,” J. Alg., 126, No. 1, 34-60 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Duchamp and D. Krob, “The lower central series of the free partially commutative group,” Semigr. Forum, 45, No. 3, 385-394 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  4. S. L. Shestakov, “The equation [x, y] = g in partially commutative groups,” Sib. Mat. Zh., 46, No. 2, 466-477 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  5. S. L. Shestakov, “The equation x 2 y 2 = g in partially commutative groups,” Sib. Mat. Zh., 47, No. 2, 463-472 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. J. Duncan, I. V. Kazachkov, and V. N. Remeslennikov, “Parabolic and quasiparabolic subgroups of free partially commutative groups,” J. Alg., 318, No. 2, 918-932 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  7. Ch. K. Gupta and E. I. Timoshenko, “Partially commutative metabelian groups: Centralizers and elementary equivalence,” Algebra Logika, 48, No. 3, 309-341 (2009).

    Article  MathSciNet  Google Scholar 

  8. E. I. Timoshenko, “Universal equivalence of partially commutative metabelian groups,” Algebra Logika, 49, No. 2, 263-290 (2010).

    Article  MathSciNet  Google Scholar 

  9. K. Hang Kim, L. Makar-Limanov, J. Neggers, and F. W. Roush, “Graph algebras,” J. Alg., 64, 46-51 (1980).

    Article  MATH  Google Scholar 

  10. L. A. Bokut’ and L.-S. Shiao, “Gröbner–Shirshov bases for Coxeter groups,” Comm. Alg., 29, No. 9, 4305-4319 (2001).

    Google Scholar 

  11. G. Duchamp and D. Krob, “The free partially commutative Lie algebra: Bases and ranks,” Adv. Math., 92, No. 1, 95-126 (1992).

    MathSciNet  Google Scholar 

  12. A. I. Shirshov, “Subalgebras of free Lie algebras,” Mat. Sb., 33(75), No. 2, 441-452 (1953).

    Google Scholar 

  13. K. T. Chen, R. H. Fox, and R. C. Lyndon, “Free differential calculus. IV: The quotient groups of the lower central series,” Ann. Math. (2), 68, 81-95 (1958).

    Google Scholar 

  14. A. I. Shirshov, “On free Lie rings,” Mat. Sb., 45(87), No. 2, 113-122 (1958).

    Google Scholar 

  15. A. I. Shirshov, “Some algorithmic problems for Lie algebras,” Sib. Mat. Zh., 3, No. 2, 292-296 (1962).

    MATH  Google Scholar 

  16. L. A. Bokut’, “Undecidability of the word problem and subalgebras of finitely presented Lie algebras,” Izv. Akad. Nauk SSSR, Ser. Mat., 36, No. 6, 1173-1219 (1972).

    MathSciNet  Google Scholar 

  17. L. A. Bokut’, Y. Fong, W.-F. Ke, and P. S. Kolesnikov, “Gröbner and Gröbner–Shirshov bases in algebra and conformal algebras,” Fund. Prikl. Mat., 6, No. 3, 669-706 (2000).

    MathSciNet  Google Scholar 

  18. L. A. Bokut’ and Y. Chen, “Gröbner–Shirshov bases for Lie algebras: After A. I. Shirshov,” Preprint, arXiv:math.RA/08041254v1.

  19. A. I. Shirshov, Selected Works, Birkhäuser (2009).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to E. N. Poroshenko.

Additional information

Translated from Algebra i Logika, Vol. 50, No. 5, pp. 595-614, September-October, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Poroshenko, E.N. Bases for partially commutative Lie algebras. Algebra Logic 50, 405–417 (2011). https://doi.org/10.1007/s10469-011-9152-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10469-011-9152-7

Keywords

Navigation