Skip to main content
SpringerLink
Log in

Coabelian Ideals in ℕ-graded Lie algebras and applications to right angled Artin Lie algebras

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We consider homological finiteness properties FPn of certain ℕ-graded Lie algebras. After proving some general results (see Theorem A, Corollary B and Corollary C), we concentrate on a family that can be considered as the Lie algebra version of the generalized Bestvina-Brady groups associated to a graph Γ. We prove that the homological finiteness properties of these Lie algebras can be determined in terms of the graph in the same way as in the group case.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. Baumslag, Free Lie algebras and free groups, Journal of the London Mathematical Society 4 (1971/72), 523–532.

    MathSciNet  MATH  Google Scholar 

  2. D. J. Benson, Representations and Cohomology. II, Cambridge Studies in Advanced Mathematics, Vol. 31, Cambridge University Press, Cambridge, 1991.

    Book  Google Scholar 

  3. M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Inventiones mathematicae 129 (1997), 445–470.

    Article  MathSciNet  Google Scholar 

  4. R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College Mathematical Notes, Queen Mary College Department of Pure Mathematics, London, 1981.

    MATH  Google Scholar 

  5. R. Bieri and B. Renz, Valuations on free resolutions and higher geometric invariants of groups, Commentarii Mathematici Helvetici 63 (1988), 464–497.

    Article  MathSciNet  Google Scholar 

  6. R. Blasco-García, J. I. Cogolludo-Agustín and C, Martínez-Pérez, On the Sigma invariants of even Artin groups of FC-type, Journal of Pure and Applied Algenra, to appear, https://arxiv.org/abs/2011.07608.

  7. W. Dicks and I. J. Leary, Presentations for subgroups of Artin groups, Proceedings of the American Mathematiucal Society 127 (1999), 343–348.

    Article  MathSciNet  Google Scholar 

  8. G. Duchamp and D. Krob, The free partially commutative Lie algebra: bases and ranks, Advances in Mathematics 95 (1992), 92–126.

    Article  MathSciNet  Google Scholar 

  9. G. Duchamp and D. Krob, The lower central series of the free partially commutative group, Semigroup Forum 45 (1992), 385–394.

    Article  MathSciNet  Google Scholar 

  10. J. D. King, Homological finiteness conditions for pro-p groups. Communications in Algebra 27 (1999), 4969–4991.

    Article  MathSciNet  Google Scholar 

  11. D. Kochloukova and C. Martínez-Pérez, Bass-Serre theory for Lie algebras: A homological approach, Journal of Algebra 585 (2021), 143–175.

    Article  MathSciNet  Google Scholar 

  12. I. Leary and M. Saadetoglu, The cohomology of Bestvina-Brady groups, Groups, Geometry and Dynamics 5 (2011), 121–138.

    Article  MathSciNet  Google Scholar 

  13. H. van der Lek, The homotopy type of complex hyperplane complements, Ph.D. thesis, University of Nijmegen, Nijmegen, 1983.

    Google Scholar 

  14. A. I. Lichtman and M. Shirvani, HNN-extensions for Lie algebras, Proceedings of the American Mathematical Society 125 (1997), 3501–3508.

    Article  MathSciNet  Google Scholar 

  15. J. Meier and L. VanWyk, The Bieri-Neuman-Strebel invariants for graph groups, Proceedings of the London Mathematical Society 71 (1995), 263–280.

    Article  MathSciNet  Google Scholar 

  16. J. Meier, H. Meinert and L. VanWyk, Higher generation subgroup sets and the Σ-invariants of graph groups, Commentarii Mathematici Helvetici 73 (1998), 22–44.

    Article  MathSciNet  Google Scholar 

  17. S. Papadima and A. Suciu, Algebraic invariants for right-angled Artin groups, Mathematische Annalen 334 (2006), 533–555.

    Article  MathSciNet  Google Scholar 

  18. S. Papadima and A. Suciu, Algebraic invariants for Bestvina-Brady groups, Journal of the London Mathematical Society 76 (2007), 273–292.

    Article  MathSciNet  Google Scholar 

  19. S. Papadima and A. Suciu, Toric complexes and Artin kernels, Advances in Mathematics 220 (2009), 441–477.

    Article  MathSciNet  Google Scholar 

  20. R. D. Wade, The lower central series of a right-angled Artin group, L’Enseignement Mathématique 61 (2015), 343–371.

    Article  MathSciNet  Google Scholar 

  21. A. Wasserman, A derivation HNN construction for Lie algebras, Israel Journal of Mathematics 106 (1998), 79–92.

    Article  MathSciNet  Google Scholar 

  22. T. Weigel, Graded Lie algebras of type FP, Israel Journal of Mathematics 205 (2015), 185–209.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

During the preparation of this work the first named author was partially supported by CNPq grant 301779/2017-1 and by FAPESP grant 2018/23690-6. The second named author was partially supported by PGC2018-101179-B-I00 and by Grupo Álgebra y Geometría, Gobierno de Aragón and Feder 2014–2020 “Construyendo Europa desde Aragón”.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Campinas, 13083 - 859, Campinas, SP, Brazil

    Dessislava H. Kochloukova

  2. Departamento de Matemáticas, Universidad de Zaragoza, 50009, Zaragoza, Spain

    Conchita Martínez-Pérez

Authors
  1. Dessislava H. Kochloukova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Conchita Martínez-Pérez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Conchita Martínez-Pérez.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kochloukova, D.H., Martínez-Pérez, C. Coabelian Ideals in ℕ-graded Lie algebras and applications to right angled Artin Lie algebras. Isr. J. Math. 247, 797–829 (2022). https://doi.org/10.1007/s11856-021-2281-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2281-3

Search

Navigation