Geodesic
On the variety of 3-dimensional~Lie algebras
Lobachevskii journal of mathematics, Tome 3 (1999), pp. 5-17.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is known that a 3-dimensional Lie algebra is unimodular or solvable as a result of the classification. We give a simple proof of this fact, based on a fundamental identity for 3-dimensiona Lie algebras, which was first appeared in [21]. We also give a representation theoretic meaning of the invariant of 3-dimensional Lie algebras introduced in [15], [22], by calculating the $GL(V)$-irreducible decomposition of polynomials on the space $\wedge^2V^*\otimes V$ up to degree 3. Typical four covariants naturally appear in this decomposition, and we show that the isomorphism classes of 3-dimensional Lie algebras are completely determined by the $GL(V)$-invariant concepts in $\wedge^2V^*\otimes V$ defined by these four covariants. We also exhibit an explicit algorithm to distinguish them. 
@article{LJM_1999_3_a0,
     author = {Y. Agaoka},
     title = {On the variety of {3-dimensional~Lie} algebras},
     journal = {Lobachevskii journal of mathematics},
     pages = {5--17},
     publisher = {mathdoc},
     volume = {3},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_1999_3_a0/}
}
TY  - JOUR
AU  - Y. Agaoka
TI  - On the variety of 3-dimensional~Lie algebras
JO  - Lobachevskii journal of mathematics
PY  - 1999
SP  - 5
EP  - 17
VL  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/LJM_1999_3_a0/
LA  - en
ID  - LJM_1999_3_a0
ER  - 
%0 Journal Article
%A Y. Agaoka
%T On the variety of 3-dimensional~Lie algebras
%J Lobachevskii journal of mathematics
%D 1999
%P 5-17
%V 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/LJM_1999_3_a0/
%G en
%F LJM_1999_3_a0
Y. Agaoka. On the variety of 3-dimensional~Lie algebras. Lobachevskii journal of mathematics, Tome 3 (1999), pp. 5-17. http://geodesic.mathdoc.fr/item/LJM_1999_3_a0/