Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

Abstract

We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

Keywords

• Lie algebra
• linear bundle of Lie algebras
• quasisimple orthogonal algebra
• Poisson manifold
• bi-Hamiltonian structure