Geodesic
On the intersection of irreducible components of the space of finite-dimensional Lie algebras
Sbornik. Mathematics, Tome 203 (2012) no. 7, pp. 976-995 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The irreducible components of the space of $n$-dimensional Lie algebras are investigated. The properties of Lie algebras belonging to the intersection of all the irreducible components of this kind are studied (these Lie algebras are said to be basic or founding Lie algebras). It is proved that all Lie algebras of this kind are nilpotent and each of these Lie algebras has an Abelian ideal of codimension one. Specific examples of founding Lie algebras of arbitrary dimension are described and, to describe the Lie algebras in general, we state a conjecture. The concept of spectrum of a Lie algebra is considered and some of the most elementary properties of the spectrum are studied. Bibliography: 6 titles.
Keywords: Lie algebra, space of Lie algebras, irreducible component, nilpotent Lie algebra. 
@article{SM_2012_203_7_a2,
     author = {V. V. Gorbatsevich},
     title = {On the intersection of irreducible components of the space of finite-dimensional {Lie} algebras},
     journal = {Sbornik. Mathematics},
     pages = {976--995},
     year = {2012},
     volume = {203},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2012_203_7_a2/}
}
TY  - JOUR
AU  - V. V. Gorbatsevich
TI  - On the intersection of irreducible components of the space of finite-dimensional Lie algebras
JO  - Sbornik. Mathematics
PY  - 2012
SP  - 976
EP  - 995
VL  - 203
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/SM_2012_203_7_a2/
LA  - en
ID  - SM_2012_203_7_a2
ER  - 
%0 Journal Article
%A V. V. Gorbatsevich
%T On the intersection of irreducible components of the space of finite-dimensional Lie algebras
%J Sbornik. Mathematics
%D 2012
%P 976-995
%V 203
%N 7
%U http://geodesic.mathdoc.fr/item/SM_2012_203_7_a2/
%G en
%F SM_2012_203_7_a2
V. V. Gorbatsevich. On the intersection of irreducible components of the space of finite-dimensional Lie algebras. Sbornik. Mathematics, Tome 203 (2012) no. 7, pp. 976-995. http://geodesic.mathdoc.fr/item/SM_2012_203_7_a2/

[1] V. V. Gorbatsevich, “Some properties of the space of $n$-dimensional Lie algebras”, Sb. Math., 200:2 (2009), 185–213 | DOI | MR | Zbl

[2] A. L. Onishchik, E. B. Vinberg, V. V. Gorbatsevich, Lie groups and Lie algebras III. Structure of Lie groups and Lie algebras, Encyclopaedia Math. Sci., 41, Springer-Verlag, Berlin, 1994 | MR | MR | Zbl | Zbl

[3] J. M. Ancochea Bermudez, M. Goze, “On the nonrationality of rigid Lie algebras”, Proc. Amer. Math. Soc., 127:9 (1999), 2611–2618 | DOI | MR | Zbl

[4] H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspects Math., D1, Vieweg, Braunschweig, 1984 | MR | MR | Zbl | Zbl

[5] V. V. Gorbatsevich, “On algebraic Anosov diffeomorphisms on nilmanifolds”, Siberian Math. J., 45:5 (2004), 821–839 | DOI | MR | Zbl

[6] R. Richardson, “On the rigidity of semi-direct products of Lie algebras”, Pacific J. Math., 22:2 (1967), 339–344 | MR | Zbl