The classification of three-dimensional Lie algebras on complex field

Abstract

In this paper, we study the classification of three-dimensional Lie al­gebras over a field of complex numbers up to isomorphism. The proposed classification is based on the consideration of objects invariant with re­spect to isomorphism, namely such quantities as the derivative of a subal­gebra and the center of a Lie algebra. The above classification is distin­guished from others by a more detailed and simple presentation. 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. Six classes of three-dimensional Lie algebras not isomorphic to each other over a field of complex numbers are presented. In each of the classes, its properties are described, as well as structural equations defining each of the Lie alge­bras. One of the reasons for considering these low dimensional Lie alge­bras that they often occur as subalgebras of large Lie algebras

Keywords

• lie algebras
• complex field
• heisenberg algebra