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The variety of Lie algebras of maximal class

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Abstract

We present an explicit description of the affine variety M Fil of Lie algebras of maximal class (filiform Lie algebras): the formulas of polynomial equations that determine this variety are written down. The affine variety M Fil can be considered as the base of the nilpotent versal deformation of the ℕ-graded Lie algebra m0.

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References

  1. D. V. Millionshchikov, “Cohomology of Graded Lie Algebras of Maximal Class with Coefficients in the Adjoint Representation,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 263, 106–119 (2008) [Proc. Steklov Inst. Math. 263, 99–111 (2008)].

    Google Scholar 

  2. A. Fialowski, “Classification of Graded Lie Algebras with Two Generators,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 2, 62–64 (1983) [Moscow Univ. Math. Bull. 38 (2), 76–79 (1983)].

  3. M. Hall, Jr., Combinatorial Theory (Blaisdell Publ. Co., Waltham, MA, 1967).

    MATH  Google Scholar 

  4. F. G. Echarte, M. C. Márquez, and J. Núñez, “A Constructive Method to Determine the Variety of Filiform Lie Algebras,” Czech. Math. J. 56, 1281–1299 (2006).

    Article  MATH  Google Scholar 

  5. A. Fialowski and D. Fuchs, “Construction of Miniversal Deformations of Lie Algebras,” J. Funct. Anal. 161(1), 76–110 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Fialowski and D. Millionschikov, “Cohomology of Graded Lie Algebras of Maximal Class,” J. Algebra 296(1), 157–176 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Fialowski and F. Wagemann, “Cohomology and Deformations of the Infinite-Dimensional Filiform Lie Algebra m0,” J. Algebra 318(2), 1002–1026 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  8. You. B. Hakimjanov, “Variété des lois d’algèbres de Lie nilpotentes,” Geom. Dedicata 40, 269–295 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  9. Yu. Khakimdjanov, “Varieties of Lie Algebra Laws,” in Handbook of Algebra, Ed. by M. Hazewinkel (North-Holland, Amsterdam, 2000), Vol. 2, pp. 509–541.

    Google Scholar 

  10. D. V. Millionschikov, “Graded Filiform Lie Algebras and Symplectic Nilmanifolds,” in Geometry, Topology, and Mathematical Physics (Am. Math. Soc., Providence, RI, 2004), AMS Transl., Ser. 2, 212, pp. 259–279.

    Google Scholar 

  11. A. Nijenhuis and R. W. Richardson, Jr., “Deformations of Lie Algebra Structures,” J. Math. Mech. 17(1), 89–105 (1967).

    MATH  MathSciNet  Google Scholar 

  12. A. Shalev and E. I. Zelmanov, “Narrow Lie Algebras: A Coclass Theory and a Characterization of the Witt Algebra,” J. Algebra 189(2), 294–331 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Shalev and E. I. Zelmanov, “Narrow Algebras and Groups,” J. Math. Sci. 93(6), 951–963 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970).

    MATH  MathSciNet  Google Scholar 

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Correspondence to Dmitry V. Millionshchikov.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 266, pp. 184–201.

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Millionshchikov, D.V. The variety of Lie algebras of maximal class. Proc. Steklov Inst. Math. 266, 177–194 (2009). https://doi.org/10.1134/S0081543809030109

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