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.

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

The existence of a class of smooth Lie algebra bundles is proved and examples are given. Further we construct smooth vector bundles which have the structure of non trivial smooth Lie algebra bundles.

On smooth lie algebra bundles

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On smooth lie algebra bundles