Low-Dimensional Lie Algebras

Abstract

We would like to know how many essentially different (that is, non-isomorphic) Lie algebras there are and what approaches we can use to classify them. To get some feeling for these questions, we shall look at Lie algebras of dimensions 1, 2, and 3. Another reason for looking at these low-dimensional Lie algebras is that they often occur as subalgebras of the larger Lie algebras we shall meet later.

Abelian Lie algebras are easily understood: For any natural number n, there is an abelian Lie algebra of dimension n (where for any two elements, the Lie bracket is zero). We saw in Exercise 1.11 that any two abelian Lie algebras of the same dimension over the same field are isomorphic, so we understand them completely, and from now on we shall only consider non-abelian Lie algebras.

How can we get going? We know that Lie algebras of different dimensions cannot be isomorphic. Moreover, if L is a non-abelian Lie algebra, then its derived algebra L′ is non-zero and its centre Z(L) is a proper ideal. By Exercise 2.8, derived algebras and centres are preserved under isomorphism, so it seems reasonable to use the dimension of L and properties of L′ and Z(L) as criteria to organise our search.