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Tr. Mat. Inst. Steklova, 2019, Volume 305, Pages 225–249 (Mi tm4011)  

Geometry of Central Extensions of Nilpotent Lie Algebras

D. V. Millionshchikova, R. Jimenezb

a Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia
b National Autonomous University of Mexico, Mexico City, 04510 Mexico

Abstract: We obtain a recurrent and monotone method for constructing and classifying nilpotent Lie algebras by means of successive central extensions. The method consists in calculating the second cohomology $H^2(\mathfrak g,\mathbb K)$ of an extendable nilpotent Lie algebra $\mathfrak g$ followed by studying the geometry of the orbit space of the action of the automorphism group $\mathrm {Aut}(\mathfrak g)$ on Grassmannians of the form $\mathrm {Gr}(m,H^2(\mathfrak g,\mathbb K))$. In this case, it is necessary to take into account the filtered cohomology structure with respect to the ideals of the lower central series: a cocycle defining a central extension must have maximum filtration. Such a geometric method allows us to classify nilpotent Lie algebras of small dimensions, as well as to classify narrow naturally graded Lie algebras. We introduce the concept of a rigid central extension and construct examples of rigid and nonrigid central extensions.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-51-55017
Direccion General de Asuntos del Personal Academico, Universidad Nacional Autonoma de Mexico
Universidad Nacional Autónoma de México
CONACYT - Consejo Nacional de Ciencia y Tecnología
The first author was supported by the Russian Foundation for Basic Research, project no. 16-51-55017. The second author was partially supported by PASPA-DGAPA, UNAM, and CONACyT.


DOI: https://doi.org/10.4213/tm4011

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English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 305, 209–231

Bibliographic databases:

UDC: 512.812.4
Received: February 3, 2019
Revised: March 4, 2019
Accepted: March 14, 2019

Citation: D. V. Millionshchikov, R. Jimenez, “Geometry of Central Extensions of Nilpotent Lie Algebras”, Algebraic topology, combinatorics, and mathematical physics, Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday, Tr. Mat. Inst. Steklova, 305, Steklov Math. Inst. RAS, Moscow, 2019, 225–249; Proc. Steklov Inst. Math., 305 (2019), 209–231

Citation in format AMSBIB
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\paper Geometry of Central Extensions of Nilpotent Lie Algebras
\inbook Algebraic topology, combinatorics, and mathematical physics
\bookinfo Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday
\serial Tr. Mat. Inst. Steklova
\yr 2019
\vol 305
\pages 225--249
\publ Steklov Math. Inst. RAS
\publaddr Moscow
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