Hopf algebra

In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map S:H ightarrow H such that the following diagram commutes

Related Topics:
Abstract algebra - Bialgebra - Field - ''K''-linear - Commutes

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Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. In the sumless Sweedler notation, this property can also be expressed as

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:S(c_{(1)})c_{(2)}=c_{(1)}S(c_{(2)})=epsilon(c)1qquadmbox{ for all }cin C.

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The map S is called the antipode map of the Hopf algebra.

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