Lie Algebra - Imported Knowlpedia Content

A Lie algebra is a vector space $\mathfrak{g}$ (typically over $\mathbb{R}$ or $\mathbb{C}$ ) equipped with a bilinear map

[\ ,\ ]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g},

called the Lie bracket, such that for all $X,Y,Z\in\mathfrak{g}$ :

[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.

Examples

Any associative algebra becomes a Lie algebra with commutator $[A,B]=AB-BA$ , e.g. matrix Lie algebras $\mathfrak{gl}(n,\mathbb{R})$ .
The space of vector fields on a manifold with the commutator bracket.
An abelian Lie algebra is one with $[X,Y]=0$ for all $X,Y$ .

Maps and structure

A structure-preserving map is a Lie algebra homomorphism; bijective ones are isomorphisms.

Important substructures include Lie subalgebras, ideals, and the center.

Many classification notions are defined in terms of the bracket, such as solvable, nilpotent, semisimple, simple, and reductive Lie algebras.