Liealgebra

A Lie algebra is a vector space g over a field F (commonly the real or complex numbers) equipped with a bilinear operation, called the Lie bracket, [ , ]: g × g → g. The bracket is antisymmetric, [x,y] = -[y,x], and satisfies the Jacobi identity, [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 for all x, y, z in g. The bracket measures the noncommutativity of an associated set of infinitesimal symmetries.

Lie algebras arise naturally from Lie groups: for a Lie group G, the tangent space at the identity together with the bracket of left-invariant vector fields forms a Lie algebra, called the Lie algebra of G. Conversely, many Lie algebras can be realized as Lie algebras of Lie groups, though the integration problem is subtle in general.

Common examples include the Lie algebra gl(n, F) of all n×n matrices with bracket [A,B] = AB − BA; the special linear algebra sl(n, F) of trace-zero matrices; and the orthogonal and symplectic algebras so(n, F) and sp(2n, F), with brackets given by the commutator.

Elements of a Lie algebra act as infinitesimal generators of continuous symmetries. A subset I ⊆ g is an ideal if [g,I] ⊆ I; ideals lead to notions of simplicity and decompositions. A Lie algebra is simple if it is nonabelian and has no nontrivial ideals. It is semisimple if it is a direct sum of simple algebras, a condition equivalent in characteristic zero to a nondegenerate Killing form.

Structure theory over C yields the classification of finite-dimensional semisimple Lie algebras via Dynkin diagrams (types A, B, C, D, and the exceptional E, F, G). The Levi decomposition states that every finite-dimensional Lie algebra over a field of characteristic zero is a semidirect sum of a semisimple subalgebra and its solvable radical.