# Hopf algebra homomorphism proofs

20.11.2019Graded Hopf algebras are often used in algebraic topology : they are the natural algebraic structure on the direct sum of all homology or cohomology groups of an H-space. Mathematics Stack Exchange works best with JavaScript enabled. Conversely, every commutative involutive reduced Hopf algebra over C with a finite Haar integral arises in this way, giving one formulation of Tannaka—Krein duality. How to proof equivalent condition of algebra morphism and coalgebra morphism about Hopf algebra Ask Question. In mathematicsa Hopf algebranamed after Heinz Hopfis a structure that is simultaneously an unital associative algebra and a counital coassociative coalgebrawith these structures' compatibility making it a bialgebraand that moreover is equipped with an antiautomorphism satisfying a certain property. A weak Hopf algebra H is usually taken to be a.

Proof. All the maps in the diagrams are algebra homomorphisms given our hy- (3) Enveloping algebras of Lie algebras: A Lie algebra g is a vector space. In terms of elements, the multiplication on B⊗B is (x⊗y)⋅(x′⊗y′)=(x⋅x′)⊗(y⋅y′)=m(x,x′)⊗m(y,y′), so in details the multiplication is. › questions › how-to-proof-equivalent-con.

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Related 1. Mathematics Stack Exchange works best with JavaScript enabled. Representative functions on a compact group. The Concise Handbook of Algebra. If a bialgebra B admits an antipode Sthen S is unique "a bialgebra admits at most 1 Hopf algebra structure".

Hopf algebra homomorphism proofs |
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Other interesting Hopf algebras are certain "deformations" or " quantizations " of those from example 3 which are neither commutative nor co-commutative. The algebra of all continuous functions on a Lie group is a locally compact quantum group. Daisy Daisy 5 5 silver badges 13 13 bronze badges. Physical Review D. |

In view of the above discussion it is . If V1 and V2 are finite dimensional this becomes an isomorphism.

(ii) The maps Δ and ε are unital algebra homomorphisms. Proof. The coassociativity of Δ follows form axiom () of a monoidal functor. The counit axiom.

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In this case:.

## Yanagihara On grouptheoretic properties of cocommutative Hopf algebras

Physical Review D. The two conditions of normality are equivalent if the antipode S is bijective, in which case A is said to be a normal Hopf subalgebra.

and let p be a Hopf algebra homomorphism of B to B'. Then the. More intrinsically, a Hopf algebra structure on an associative algebra is precisely the The antipode is an antihomomorphism both of algebras and coalgebras (i.e. a homomorphism S:A→Acopop). Proof (algebra part).

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The proof consists of writing down the four commutative diagrams corresponding to third and fourth characterizations. Asked 3 years, 7 months ago.

Turaev in are also generalizations of Hopf algebras.

Video: Hopf algebra homomorphism proofs

This is the smallest example of a Hopf algebra that is both non-commutative and non-cocommutative. The Concise Handbook of Algebra.

Hopf algebra homomorphism proofs |
Sign up or log in Sign up using Google. Video: Hopf algebra homomorphism proofs If a bialgebra B admits an antipode Sthen S is unique "a bialgebra admits at most 1 Hopf algebra structure". Post as a guest Name. Bibcode : JHEP The two conditions of normality are equivalent if the antipode S is bijective, in which case A is said to be a normal Hopf subalgebra. Furthermore, we can define the trivial representation as the base field K with. |

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The Nichols—Zoeller freeness theorem established in that the natural A -module H is free of finite rank if H is finite-dimensional: a generalization of Lagrange's theorem for subgroups.

Question feed. The axioms are partly chosen so that the category of H -modules is a rigid monoidal category.