# 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.

Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other.

### Other articles related to "hopf algebra, hopf, algebra, hopf algebras":

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 ... 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
... Groups can be axiomatized by the same diagrams (equivalently, operations) as a Hopf algebra, where G is taken to be a set instead of a module ... is the inverse In this philosophy, a group can be thought of as a Hopf algebra over the "field with one element" ...
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 isomorphic to ...
Hopf Algebras - Examples
... Counit Antipode Commutative Cocommutative Remarks group algebra KG group G Δ(g) = g ⊗ g for all g in G ε(g) = 1 for all g in G S(g) = g −1 for all g ... 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 ... This is the smallest example of a Hopf algebra that is both non-commutative and non-cocommutative ...
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 ... The notion should not be confused with quasitriangular Hopf algebra ...