Lie algebras

E542122

algebraic structure nonassociative algebra

Lie algebras are algebraic structures used to study continuous symmetries, especially those arising from Lie groups, via a linearized, infinitesimal perspective.

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Observed surface forms (2)

Surface form	Occurrences
Lie algebra	2
Lie algebra of SU(2)	1

Statements (60)

Predicate	Object
instanceOf	algebraic structure ⓘ nonassociative algebra ⓘ
appliedIn	gauge theory ⓘ particle physics ⓘ quantum mechanics ⓘ string theory NERFINISHED ⓘ
definedOver	field ⓘ
fieldOfStudy	abstract algebra ⓘ differential geometry ⓘ representation theory ⓘ theoretical physics ⓘ
generalizationOf	Lie algebra over a ring ⓘ
hasConcept	Cartan decomposition NERFINISHED ⓘ Levi decomposition NERFINISHED ⓘ center of a Lie algebra ⓘ derivation of a Lie algebra ⓘ homomorphism of Lie algebras ⓘ ideal of a Lie algebra ⓘ nilpotent Lie algebra ⓘ quotient Lie algebra ⓘ reductive Lie algebra ⓘ semisimple Lie algebra ⓘ simple Lie algebra ⓘ solvable Lie algebra ⓘ subalgebra ⓘ
hasExample	Heisenberg Lie algebra NERFINISHED ⓘ Lie algebra gl(n,F) ⓘ Lie algebra sl(n,F) ⓘ Lie algebra so(n,F) ⓘ Lie algebra sp(2n,F) ⓘ Virasoro algebra NERFINISHED ⓘ Witt algebra NERFINISHED ⓘ abelian Lie algebra ⓘ matrix Lie algebra ⓘ
hasHistoricalPeriod	late 19th century ⓘ
hasKeyResult	Ado's theorem NERFINISHED ⓘ Cartan classification of complex semisimple Lie algebras NERFINISHED ⓘ Levi–Malcev decomposition NERFINISHED ⓘ Weyl's theorem on complete reducibility NERFINISHED ⓘ
hasOperation	Lie bracket ⓘ
hasProperty	Jacobi identity NERFINISHED ⓘ alternating bracket ⓘ antisymmetric bracket ⓘ bilinear bracket ⓘ
hasStructure	vector space ⓘ
namedAfter	Sophus Lie NERFINISHED ⓘ
originatesFrom	study of Lie groups ⓘ
relatedTo	Cartan subalgebra ⓘ Killing form NERFINISHED ⓘ Lie algebra cohomology NERFINISHED ⓘ Lie algebra representation ⓘ Lie group NERFINISHED ⓘ root system ⓘ universal enveloping algebra ⓘ
satisfies	[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 ⓘ [x,x] = 0 for all x ⓘ
specialCaseOf	nonassociative algebra ⓘ
usedFor	infinitesimal symmetries ⓘ study of Lie groups ⓘ study of continuous symmetries ⓘ

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

Lie theory → fieldOfStudy → Lie algebras ⓘ

Lie ring → isRingTheoreticAnalogueOf → Lie algebras ⓘ

this entity surface form: Lie algebra

Theorie der Transformationsgruppen → mainSubject → Lie algebras ⓘ

Marius → knownFor → Lie algebras ⓘ

subject surface form: Marius Sophus Lie

Sophus → notableFor → Lie algebras ⓘ

subject surface form: Sophus Lie

Lie bracket → codomain → Lie algebras ⓘ

this entity surface form: Lie algebra

Pauli matrices → basisOf → Lie algebras ⓘ

this entity surface form: Lie algebra of SU(2)

The Poincaré-Birkhoff-Witt theorem in ring theory → mainSubject → Lie algebras ⓘ