universal enveloping algebras

E542123

algebraic construction associative algebra

Universal enveloping algebras are associative algebras that encode the structure of Lie algebras and allow Lie-theoretic problems to be studied using tools from associative and representation theory.

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Observed surface forms (2)

Surface form	Occurrences
Algèbres enveloppantes	1
universal enveloping algebra	0

Statements (49)

Predicate	Object
instanceOf	algebraic construction ⓘ associative algebra ⓘ
appliedIn	differential operators on Lie groups ⓘ harmonic analysis on Lie groups ⓘ theory of distributions on Lie groups ⓘ
associatedFunctor	left adjoint to the forgetful functor from associative algebras to Lie algebras ⓘ
associatedGradedAlgebra	symmetric algebra of the Lie algebra ⓘ
canonicalMap	Lie algebra homomorphism i: g → U(g) ⓘ
categoryCodomain	category of associative algebras ⓘ
categoryDomain	category of Lie algebras ⓘ
constructedByQuotienting	tensor algebra by the ideal generated by x⊗y − y⊗x − [x,y] ⓘ
constructedFrom	tensor algebra of the underlying vector space ⓘ
containsAsSubspace	original Lie algebra via canonical map ⓘ
definedFor	Lie algebra over a commutative ring ⓘ Lie algebra over a field NERFINISHED ⓘ
encodesStructureOf	Lie algebra ⓘ
field	mathematics ⓘ
generalizedBy	Hopf algebraic enveloping algebra ⓘ quantized universal enveloping algebra ⓘ
hasBasisDescribedBy	Poincaré–Birkhoff–Witt basis NERFINISHED ⓘ
hasProperty	associative ⓘ generally noncommutative ⓘ unital ⓘ
hasStructure	Hopf algebra ⓘ
HopfStructureIncludes	antipode ⓘ comultiplication ⓘ counit ⓘ
isFilteredBy	degree of tensors ⓘ
PBWTheoremStates	associated graded algebra is isomorphic to symmetric algebra of the Lie algebra ⓘ
relatedConcept	Lie algebra representation ⓘ associative algebra representation ⓘ symmetric algebra ⓘ tensor algebra ⓘ
satisfies	Poincaré–Birkhoff–Witt theorem NERFINISHED ⓘ universal property for Lie algebra homomorphisms into associative algebras ⓘ
specialCase	group algebra when Lie algebra is abelian and discrete analog is considered ⓘ
subfield	Lie theory ⓘ homological algebra ⓘ noncommutative algebra ⓘ representation theory ⓘ
universalProperty	every Lie algebra homomorphism into the Lie algebra of an associative algebra factors uniquely through it ⓘ
usedFor	studying representations of Lie algebras ⓘ translating Lie-theoretic problems into associative algebra problems ⓘ
usedIn	Dixmier’s work on enveloping algebras ⓘ Harish-Chandra theory NERFINISHED ⓘ classification of representations of semisimple Lie algebras ⓘ highest weight representation theory ⓘ primitive ideal theory ⓘ study of algebraic groups via Lie algebras ⓘ

Referenced by (2)

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

Lie theory → studies → universal enveloping algebras ⓘ

Jacques Dixmier → notableWork → universal enveloping algebras ⓘ

this entity surface form: Algèbres enveloppantes