Bianchi spaces and their three-dimensional isometries as S-expansions of two-dimensional isometries

Abstract

In this paper we show that certain three-dimensional isometry algebras, specifically those of type I, II, III and V (according to Bianchi’s classification), can be obtained as expansions of the isometries in two dimensions. In particular, we use the so-called S-expansion method, which makes use of the finite Abelian semigroups, because it is the most general procedure known until now. Also, it is explicitly shown why it is impossible to obtain the algebras of type IV, VI–IX as expansions from the isometry algebras in two dimensions. All the results are checked with computer programs. This procedure shows that the problem of how to relate, by an expansion, two Lie algebras of different dimensions can be entirely solved. In particular, the procedure can be generalized to higher dimensions, which could be useful for diverse physical applications, as we discuss in our conclusions.