Lie algebras

vector space with an alternating binary operation satisfying the Jacobi identity.

invariant symmetric bilinear form defined on Lie algebra; trace of composition of two adjoint representations

Property of some binary operations, such as the cross product and any ring's commutator

geometric arrangements of points, foundational to Lie theory

subgroup of a group G that each leaves invariant each element of a given subset of a G-set

complex Lie algebra, the unique central extension of the Witt algebra

set of matrices useful in studying the strong force

matrices named after Élie Cartan

subgroup of a root system's isometry group

pictoral representation of symmetry

Lie algebra defined by generators and relations through a generalized Cartan matrix

in mathematics, a family of finite groups

vertex algebra equipped with a conformal element

distinguished element of a Lie algebra's center

module over a commutative ring with a bilinear product [–, –] such that for any element x, [–, x] is a derivation

type of Lie group

nilpotent subalgebra of a Lie algebra

coefficients specifying products of basis vectors as linear combinations of other basis vectors, sufficient to construct a (non-unital) algebra over a field from any vector space

theorem stating that every finite-dimensional Lie algebra over a field is a Lie algebra of square matrices

algebraic construction of the cohomology of (the underlying manifold of) a simply connected Lie group in terms of its Lie algebra

generalization of a Lie algebra with ℤ/2 grading

Lie algebra that is not Abelian and has no nonzero proper ideals

explicitly describes the universal enveloping algebra of a Lie algebra

theorem

criterion for solvability of a Lie algebra over a characteristic-zero field

map from Lie algebra to Lie group

one of basic infinite-dimensional Lie algebras

basis of an algebraic structure that is canonical in a sense that depends on the precise context

concept in Lie-algebra representation theory

structure on Lie groups

vector space equipped with both a Lie bracket and a Lie cobracket, with certain compatibility conditions between the two

infinitesimal version of a Lie groupoid: manifold M with vector bundle E, vector bundle map ρ: E→TM, and a Lie bracket on sections of E, so that [s,ft] = (ρ(s)f)t+f[s,t] for any function f: M→ℝ and sections s, t of E

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type of Kac–Moody algebras

an analogue of the formula det(AB) = det(A) det(B), for matrices with noncommuting entrie

a finite dimensional Lie algebra is a semidirect product of a normal solvable ideal and a semisimple subalgebra

theorem in the mathematics of Lie's theory

Lists all real 3-dimensional Lie algebras