Hopf algebra
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:
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.
Examples
Group algebra
An example of a Hopf algebra is group algebra . She is going through
- For
and
- For
to a bialgebra , the antipode
- For
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 .
literature
- Christian Kassel : Quantum Groups (= Graduate Texts in Mathematics . 155). Springer, New York NY et al. 1995, ISBN 0-387-94370-6 .
- Moss E. Sweedler : Hopf algebras. Benjamin, New York NY 1969.