Lie algebra - Knowlpedia

Let $k$ be a field. A Lie algebra over $k$ is a vector space $\mathfrak g$ over $k$ equipped with a $k$ -bilinear map

[-,-]:\mathfrak g\times\mathfrak g\longrightarrow\mathfrak g,

called the Lie bracket, such that $[X,X]=0$ and

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

The first condition says that the bracket is alternating; it implies $[X,Y]=-[Y,X]$ . Over a field of characteristic different from $2$ , the two formulations are equivalent.

Base field and finite-dimensional convention

The base field is part of the structure and should be stated. A complex Lie algebra can be regarded as a real Lie algebra by restricting scalars, but its real dimension doubles and real-linear homomorphisms need not be complex-linear.

Unless a larger category is explicitly named, the finite-dimensional category $\mathbf{LieAlg}^{\mathrm{fd}}_k$ has finite-dimensional Lie algebras over $k$ as objects and Lie algebra homomorphisms as morphisms. The characteristic-zero case, especially $k=\mathbb R$ or $\mathbb C$ , is the setting for the formal Lie correspondence.

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.

References

Nicolas Bourbaki, Lie Groups and Lie Algebras: Chapters 1–3, Springer, 1989. Publisher record. Relevant: Chapter 1, Lie algebras.
Anthony W. Knapp, Lie Groups Beyond an Introduction, second edition, Birkhäuser, 2002. Publisher record. Relevant: Chapter I, Lie algebras and Lie groups.