×

The variety of Lie algebras of maximal class. (English. Russian original) Zbl 1223.17016

Proc. Steklov Inst. Math. 266, 177-194 (2009); translation from Tr. Mat. Inst. Steklova 266, 184-201 (2009).
An \(n\)-dimensional nilpotent Lie algebra is called filiform if its nilpotency class is \(n-1\). M. Vergne [Bull. Soc. Math. Fr. 98, 81–116 (1970; Zbl 0244.17011)] constructed the first known examples of filiform Lie algebras \(\mathfrak m_{0}(n)\). Vergne showed that an arbitrary filiform Lie algebra is isomorphic to a certain deformation of \(\mathfrak m_{0}(n)\) by a cocycle in the second cohomology of \(\mathfrak m_{0}(n)\) with coefficients in the adjoint representation of \(\mathfrak m_{0}(n)\). Furthermore, Vergne proved that any finite-dimensional \(\mathbb N\)-graded filiform Lie algebra is isomorphic to either \(\mathfrak m_{0}(2r)\) or \(\mathfrak m_{1}(2s+1)\), for some numbers \(r,s\), where the the structure constants of \(\mathfrak m_{0}(2r)\) and \(\mathfrak m_{1}(2s+1)\) are given. Let \(\mathfrak m_{0}\) denote the direct limit of the Lie algebras \(\mathfrak m_{0}(n)\). It turns out that \(\mathfrak m_{0}\) is an \(\mathbb N\)-graded Lie algebra of maximal class (coclass 1). The paper under review describes all infinite-dimensional Lie algebras of maximal class in terms of deformations of \(\mathfrak m_{0}\) by certain cocycles in the second cohomology of \(\mathfrak m_{0}\) with coefficients in the adjoint representation of \(\mathfrak m_{0}\).

MSC:

17B30 Solvable, nilpotent (super)algebras
17B01 Identities, free Lie (super)algebras
17B56 Cohomology of Lie (super)algebras

Citations:

Zbl 0244.17011
PDFBibTeX XMLCite
Full Text: DOI arXiv

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)]. · Zbl 1196.17018
[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)]. · Zbl 0533.17008
[3] M. Hall, Jr., Combinatorial Theory (Blaisdell Publ. Co., Waltham, MA, 1967).
[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). · Zbl 1164.17012 · doi:10.1007/s10587-006-0094-5
[5] A. Fialowski and D. Fuchs, ”Construction of Miniversal Deformations of Lie Algebras,” J. Funct. Anal. 161(1), 76–110 (1999). · Zbl 0944.17015 · doi:10.1006/jfan.1998.3349
[6] A. Fialowski and D. Millionschikov, ”Cohomology of Graded Lie Algebras of Maximal Class,” J. Algebra 296(1), 157–176 (2006). · Zbl 1147.17016 · doi:10.1016/j.jalgebra.2005.10.031
[7] A. Fialowski and F. Wagemann, ”Cohomology and Deformations of the Infinite-Dimensional Filiform Lie Algebra m0,” J. Algebra 318(2), 1002–1026 (2007). · Zbl 1181.17008 · doi:10.1016/j.jalgebra.2007.09.004
[8] You. B. Hakimjanov, ”Variété des lois d’algèbres de Lie nilpotentes,” Geom. Dedicata 40, 269–295 (1991). · doi:10.1007/BF00189914
[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. · Zbl 0974.17004
[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. · Zbl 1077.17013
[11] A. Nijenhuis and R. W. Richardson, Jr., ”Deformations of Lie Algebra Structures,” J. Math. Mech. 17(1), 89–105 (1967). · Zbl 0166.30202
[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). · Zbl 0886.17008 · doi:10.1006/jabr.1996.6819
[13] A. Shalev and E. I. Zelmanov, ”Narrow Algebras and Groups,” J. Math. Sci. 93(6), 951–963 (1999). · Zbl 0933.17011 · doi:10.1007/BF02366350
[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). · Zbl 0244.17011 · doi:10.24033/bsmf.1695
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.