Lie group

E142004

differentiable manifold group mathematical structure smooth manifold topological group

A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.

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All labels observed (4)

Label	Occurrences
Lie groups	17
Lie group canonical	4
Heisenberg group	1
Lie group actions	1

Statements (68)

Predicate	Object
instanceOf	differentiable manifold ⓘ group ⓘ mathematical structure ⓘ smooth manifold ⓘ topological group ⓘ
appearsIn	gauge theory ⓘ general relativity ⓘ particle physics ⓘ quantum mechanics ⓘ string theory ⓘ
dimension	finite-dimensional (for finite-dimensional Lie groups) ⓘ
fieldOfStudy	Lie theory ⓘ differential geometry ⓘ mathematics ⓘ representation theory ⓘ theoretical physics ⓘ
generalizationOf	continuous groups of transformations ⓘ matrix groups ⓘ
hasAssociatedObject	Cartan subgroup ⓘ Weyl group ⓘ surface form: Weyl group (for semisimple Lie groups) maximal compact subgroup ⓘ root system (for semisimple Lie groups) ⓘ universal covering group ⓘ
hasExample	Heisenberg Lie algebra ⓘ surface form: Heisenberg group circle group U(1) ⓘ complex numbers under addition ⓘ general linear group GL(n,C) ⓘ general linear group GL(n,R) ⓘ nonzero real numbers under multiplication ⓘ real numbers under addition ⓘ special linear group SL(n,C) ⓘ special linear group SL(n,R) ⓘ special orthogonal group SO(n) ⓘ special unitary group SU(n) ⓘ
hasPart	Lie algebra ⓘ
hasProperty	Hausdorff ⓘ continuous symmetries ⓘ group operation is smooth ⓘ inversion map is smooth ⓘ inversion map is smooth diffeomorphism ⓘ locally Euclidean ⓘ multiplication map is smooth ⓘ second countable ⓘ
hasType	abelian Lie group ⓘ compact Lie group ⓘ connected Lie group ⓘ finite-dimensional Lie group ⓘ infinite-dimensional Lie group ⓘ nilpotent Lie group ⓘ non-compact Lie group ⓘ reductive Lie group ⓘ semisimple Lie group ⓘ simply connected Lie group ⓘ solvable Lie group ⓘ
namedAfter	Sophus Lie ⓘ
relatedConcept	Lie algebra ⓘ Lie group action ⓘ Lie group representation ⓘ Lie homomorphism ⓘ Lie ring ⓘ Lie semigroup ⓘ Lie subgroup ⓘ
studiedBy	Sophus Lie ⓘ
studiedIn	19th century ⓘ
usedFor	classification of symmetries in physics ⓘ representation theory of groups ⓘ study of continuous symmetries ⓘ study of differential equations ⓘ