On the variety of 3-dimensional Lie algebras
(Lobachevskii Journal of Mathematics, Volume III)
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-dimensional Lie algebras, which was first appeared in . We also give a representation theoretic meaning of the invariant of 3-dimensional Lie algebras introduced in , , by calculating the GL(V)-irreducible decomposition of polynomials on the space Λ2 V∗⊗ 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 Λ2 V∗⊗ V defined by these four covariants. We also exhibit an explicit algorithm to distinguish them.