Hopf algebra

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Hopf algebra

touches the specialties

is a special case of

A Hopf algebra - named after the mathematician Heinz Hopf - over a body is a bialgebra with a linear mapping, the so-called "antipode" , so that the following diagram commutes:

Diagram defining property of the antipode

Formally written in the Sweedler notation - named after Moss Sweedler - this means:

Convolution and antipode

Be an algebra and a koalgebra . The linear mappings from to form an algebra with a product , called convolution, defined by


The neutral element in this algebra is because

and accordingly also


For a bialgebra the linear mappings from to form an algebra in this way. The antipode is the inverse element of the identical mapping in this algebra. This means


It can be shown that the antipode of a Hopf algebra is always unique, and at the same time it is an anti-algebra homomorphism and an anticoalgebra homomorphism. With the help of this fact, the value of the antipode on each element of the Hopf algebra can be calculated if the values ​​of the antipode on an algebra-generating system are known.


Group algebra

An example of a Hopf algebra is group algebra . She is going through




to a bialgebra , the antipode


turns it into a Hopf algebra.

Universal enveloping algebra

The universal enveloping algebra of a Lie algebra is naturally a Hopf algebra. For one element , the coproduct is through

and the Koeins through

Are defined.

defines the antipode.

Group-like and primitive elements

An element of a Hopf algebra is called “group-like” if and . Then applies to the antipode .

An element is called "primitive" if . It follows that and .

An element is called "skew-primitive" if with group-like elements and . It follows that and .