On the variety of 3-dimensional Lie algebras

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.