## 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)

... We have the two dimensional

*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

... In mathematics a 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 Algebra**s - 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 Algebra**s - 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" ...

### Famous quotes containing the word algebra:

“Poetry has become the higher *algebra* of metaphors.”

—José Ortega Y Gasset (1883–1955)

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