Lie Algebra

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

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

1. Alternating / antisymmetry: $[X,X]=0$ (equivalently $[X,Y]=-[Y,X]$).
2. Jacobi identity: $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.$

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.