Maximal abelian subalgebras of e(p, q) algebras Original Research Article

Maximal abclian subalgebras (MASAs) of one of the classical real inhomogencous Lie algebras are constructed, namely those of the pseudoeuclidean Lie algebra e(p, q). Use is made of the semidireet sum structure of e(p,q) with the translations T(p + q) as an abclian ideal. We first construct splitting MASAs that arc themselves direct sums of abelian subalgebras of o(p,q) and of subalgebras of T(p + q). The splitting subalgebras are used to construct the complementary nonsplitting ones. Here the results are less complete than in the splitting case. We present general decomposition theorems and construct indecomposable MASAs for all algebras e(p,q), p ≥ q ≥ 0. The case of q = 0 and 1 were treated earlier in a physical context. The case q = 2 is analyzed here in detail as an illustration of the general results.