Lie algebra
Noun
Meaning
Lie algebra (plural Lie algebras)
(mathematics) A linear algebra whose mathematical structure underlies a Lie group’s structure.
Examples
A basis, then, is a set of generators being a basis of the Lie algebra in the usual vector space sense.
Ado's theorem says every finite-dimensional Lie algebra is isomorphic to a matrix Lie algebra.
A group parameter is a component of a coordinate vector representing an arbitrary element of the Lie algebra with respect to some basis.
A Lie algebra is solvable if and only if Classification The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way.
As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup.
Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.