Yoshio Agaoka
On the variety of 3-dimensional Lie algebras
(Lobachevskii Journal of Mathematics, Vol.3, pp.5-17)
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.
We also give a representation theoretic meaning of the invariant of 3-dimensional
Lie algebras by calculating the $GL(V)$-irreducible decomposition of polynomials
on the space $\wedge^{2} V^{*}\otimes V$ 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 $\wedge^{2} V^{*}\otimes V$ defined by these
four covariants. We also exhibit an explicit algorithm to distinguish them.
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