# What is hopf algebra?

## Hopf Algebra

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

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### Some articles on hopf algebra:

Supersymmetry As A Quantum Group - Unitary (-1)F Operator
... We have the two dimensional Hopf algebra generated by (-1)F subject to with the counit and the coproduct and the antipode Thus far, there is nothing supersymmetric about this Hopf algebra at all it is ...
Braided Hopf Algebra
... In mathematics a braided Hopf algebra is a Hopf algebra in a braided monoidal category ... The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra H, particurlarly the Nichols algebra of a braided vectorspace in that ... The notion should not be confused with quasitriangular Hopf algebra ...
Hopf Algebras - Examples
... Depending on Comultiplication Counit Antipode Commutative Cocommutative Remarks group algebra KG group G Δ(g) = g ⊗ g for all g in G ε(g) = 1 ... Tensor algebra T(V) vector space V Δ(x) = x ⊗ 1 + 1 ⊗ x, x in V ε(x) = 0 S(x) = -x for all x in T1(V) (and extended to higher tensor powers) no yes symmetric algebra and exterior algebra (whi ... This is the smallest example of a Hopf algebra that is both non-commutative and non-cocommutative ...
Hopf Algebras - Formal Definition - Hopf Subalgebras
... A subalgebra A of a Hopf algebra H is a Hopf subalgebra if it is a subcoalgebra of H and the antipode S maps A into A ... In other words, a Hopf subalgebra A is a Hopf algebra in its own right when the multiplication, comultiplication, counit and antipode of H is restricted to A (and additionally the identity 1 of ... As a corollary of this and integral theory, a Hopf subalgebra of a semisimple finite-dimensional Hopf algebra is automatically semisimple ...
Hopf Algebra - Analogy With Groups
... same diagrams (equivalently, operations) as a Hopf algebra, where G is taken to be a set instead of a module ... a group can be thought of as a Hopf algebra over the "field with one element" ...