Maurer–Cartan form

E542126

Lie algebra-valued 1-form differential 1-form geometric structure

The Maurer–Cartan form is a canonical Lie algebra-valued 1-form on a Lie group that encodes its infinitesimal structure and underlies many constructions in differential geometry and gauge theory.

Try in SPARQL Jump to: Statements Referenced by

Statements (47)

Predicate	Object
instanceOf	Lie algebra-valued 1-form ⓘ differential 1-form ⓘ geometric structure ⓘ
appearsIn	BRST formalism NERFINISHED ⓘ L_ ∞-algebra theory ⓘ Wess–Zumino–Witten models NERFINISHED ⓘ definition of Cartan connection ⓘ deformation theory ⓘ integrability conditions for G-structures ⓘ nonlinear sigma models in theoretical physics ⓘ theory of principal bundles ⓘ
codomainIncludes	tensor product of cotangent bundle with Lie algebra ⓘ
definedAs	g^{-1}dg for matrix Lie groups ⓘ
definedOn	Lie group ⓘ
determines	Lie bracket on the Lie algebra ⓘ
domainIncludes	every point of the Lie group ⓘ
encodes	infinitesimal structure of a Lie group ⓘ
generalizes	logarithmic derivative on Lie groups ⓘ
is	canonical ⓘ flat connection form on a principal bundle over the Lie group ⓘ invariant under right action up to adjoint action ⓘ left-invariant ⓘ
isCharacterizedBy	being identity on the Lie algebra at the identity element ⓘ left-translation invariance ⓘ
isToolFor	constructing representations of Lie groups ⓘ describing gauge fields as connection 1-forms ⓘ studying local properties of Lie groups ⓘ
namedAfter	Ludwig Maurer NERFINISHED ⓘ Élie Cartan NERFINISHED ⓘ
pullbackBy	left translation on the Lie group ⓘ
relatedTo	exponential map of a Lie group ⓘ holonomy of flat connections ⓘ structure constants of a Lie algebra ⓘ
satisfies	Maurer–Cartan equation NERFINISHED ⓘ
takesValuesIn	Lie algebra ⓘ
transformsBy	adjoint representation of the Lie group ⓘ
usedIn	Cartan geometry NERFINISHED ⓘ Lie algebra cohomology ⓘ Lie group theory NERFINISHED ⓘ connection theory ⓘ differential geometry ⓘ gauge theory ⓘ
usedToDefine	canonical symplectic form on cotangent bundle of a Lie group ⓘ left-invariant vector fields ⓘ right-invariant vector fields ⓘ
usedToExpress	curvature of connections ⓘ structure equations of Cartan ⓘ

Referenced by (1)

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

Cartan connections → influencedBy → Maurer–Cartan form ⓘ

subject surface form: Cartan connection