Varieties of nilpotent Lie algebras of dimension less than six (Q1099239)

The set of all nilpotent Lie algebra multiplications on a fixed vector space of dimension \(n\) over a field \(k\) forms an affine variety \({\mathfrak N}_ n\). Isomorphism classes of these Lie algebras are orbits in \({\mathfrak N}_ n\) under the ``change of basis'' action of \(\text{GL}(n,k)\). By examining the orbit structure of \({\mathfrak N}_ n\) for small \(n\) the author gives a new perspective on the classification of nilpotent Lie algebras.