## Yoshio Agaoka

# 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 [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 *Λ*^{2} 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 *Λ*^{2} V^{∗}⊗ V defined by these four covariants. We also exhibit an explicit algorithm to distinguish them.