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Computing the index of Lie algebras; pp. 265–271

Full article in PDF format | doi: 10.3176/proc.2010.4.03

Hadjer Adimi, Abdenacer Makhlouf


The aim of this paper is to compute and discuss the index of Lie algebras. We consider the n-dimensional Lie algebras for n < 5 and the case of filiform Lie algebras which form a special class of nilpotent Lie algebras. We compute the index of generalized Heisenberg algebras and graded filiform Lie algebras Ln and Qn. We also discuss the evolution of the Lie algebra index by deformation.


  1. Adimi, H. Sur les indices des algèbres de Lie et algèbres associatives. Master’s thesis, Sétif University, 2005.

  2. Goze, M. and Ancochea-Bermudez, J.-M. On the varieties of nilpotent Lie algebras of dimension 7 and 8. J. Pure Appl. Algebra, 1992, 77, 131–140.

  3. Dergachev, F. and Kirillov, A. Index of Lie algebras of seaweed type. J. Lie Theory, 2000, 10, 331–343.

  4. Dixmier, J. Enveloping Algebras. Graduate Studies in Mathematics, Vol. 11, AMS, 1996.

  5. Elashvili, A. G. On the index of parabolic subalgebras of semisimple Lie algebras. Preprint, 1990.

  6. Elashvili, A. G. On the index of orisphorical subalgebras of semisimple Lie algebras. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 1985, 77, 116–126.

  7. Elashvili, A. G. Frobenius Lie algebras. Funksional. Anal. i Prilozhen., 1982, 16, 94–95. English translation: Functional Anal. Appl., 1982, 16, 4 (1983), 326–328.

  8. Elashvili, A. G. Frobenius Lie algebras II. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 1985, 77, 127–137.

  9. Gerstenhaber, M. On the deformation of rings and algebras. Ann. Math., 1964, 79, 59–103.

10. Gerstenhaber, M. On the deformation of rings and algebras II. Ann. Math., 1966, 84, 1–19.

11. Gerstenhaber, M. On the deformation of rings and algebras III. Ann. Math., 1968, 88, 1–34.

12. Gerstenhaber, M. On the deformation of rings and algebras IV. Ann. Math., 1974, 99, 257–276.

13. Gòmez, J. R., Jimenéz-Merchán, A., and Khakimdjanov, Y. Low-dimensional filiform Lie algebras. J. Pure Appl. Algebra, 1998, 130, 133–158.

14. Goze, M. and Khakimdjanov, Y. Nilpotent Lie Algebras. Kluwer Academic Publishers, MIA 361, 1996.

15. Makhlouf, A. Comparison of deformations and geometric study of associative algebras varieties. Int. J. Math. Math. Sci., 2007, Article ID 18915, 24 pages.

16. Ooms, A. I. On Frobenius algebras. Commun. Algebra, 1980, 8, 13–52.

17. Ooms, A. I. Computing invariants and semi-invariants by means of Frobenius Lie algebras. 2008, arXiv:0806.4178 [math.RT].

18. Raïs, M. and Tauvel, P. Indice et polynômes invariants pour certaines algèbres de Lie. J. Reine Angew. Math., 1992, 425, 123–140.

19. Seeley, C. 7-dimensional nilpotent Lie algebras. Trans. Amer. Math. Soc., 1993, 335, 479–496.

20. Tauvel, P. and Yu, R. W. T. Indice et formes linéaires stables dans les algèbres de Lie. J. Algebra, 2004, 273, 507–516.

21. Tauvel, P. and Yu, R. W. T. Sur l’indice de certaines algèbres de Lie. Ann. Inst. Fourier (Grenoble), 2004, 54, 1793–1810.

22. Vergne, M. Cohomologie des algèbres de Lie nilpotentes. Application á l’étude de la variété des algèbres de Lie nilpotentes. Bull. Soc. Math. France, 1970, 98, 81–116.

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