Hopf algebra (concept named after him)

E679321

A Hopf algebra is an abstract algebraic structure that unifies and generalizes groups, rings, and vector spaces, playing a central role in areas such as algebraic topology, quantum groups, and category theory.

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Hopf algebra (concept named after him) canonical 1

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Predicate Object
instanceOf algebraic structure
mathematical concept
appearsIn algebraic K-theory NERFINISHED
stable homotopy theory
condition antipode is convolution inverse of identity
comultiplication is an algebra homomorphism
counit is an algebra homomorphism
definedOver commutative ring
field
example Connes–Kreimer Hopf algebra NERFINISHED
Hopf algebra of symmetric functions
Sweedler’s 4-dimensional Hopf algebra NERFINISHED
coordinate ring of an algebraic group
group algebra of a group
quantum enveloping algebra U_q(g)
universal enveloping algebra of a Lie algebra
field abstract algebra
algebraic topology
category theory NERFINISHED
noncommutative geometry
quantum algebra
representation theory
generalizes Lie algebra (via universal enveloping algebra) NERFINISHED
bialgebra
coalgebra
group
group algebra
ring
hasPart algebra structure
antipode
coalgebra structure
comultiplication
counit
multiplication
unit
namedAfter Heinz Hopf NERFINISHED
property algebra and coalgebra structures are compatible
antipode is anticomultiplicative
antipode is antimultiplicative
comultiplication is coassociative
counit satisfies counit axioms
simultaneously an algebra and a coalgebra
relatedConcept Tannaka–Krein duality NERFINISHED
bialgebra
monoidal category
quantum group
usedIn Hopf–Galois theory NERFINISHED
algebraic topology
combinatorics
deformation quantization
homotopy theory
knot invariants
quantum groups
topological quantum field theory

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Heinz Hopf notableWork Hopf algebra (concept named after him)