Hopf-Algebra
Prejudice to the special areas
- Mathematics Abstract Algebra
- Linear Algebra
- Commutative Algebra
Is a special case of
A Hopf algebra - named after the mathematician Heinz Hopf - over a field is a Bialgebra with a - linear map, called the " antipode ", so that the following diagram commutes:
Formally, in the Sweedler notation - named after Moss Sweedler - written means:
Folding and antipode
Be an algebra and a coalgebra. The - linear maps from to form an algebra with product, called convolution, defined by
The neutral element in this algebra is because
And correspondingly
For a Bialgebra form - linear maps from to in this way an algebra. The antipode is the inverse to the identity mapping element in this algebra. that is
It can be shown that the antipode of a Hopf algebra is always unique, and at the same time is a Antialgebrahomomorphismus and a Anticoalgebrahomomorphismus. Using this fact can calculate the value of the antipode on each element of the Hopf algebra, if the values of the antipode are known on a algebra of generators.
Examples
Group algebra
A simple example of a Hopf algebra is the group algebra. It is carried
And
To a Bialgebra, the antipode
Makes it a Hopf algebra.
Universal enveloping algebra
The universal enveloping algebra of a Lie algebra is naturally a Hopf algebra. For an element of the coproduct is by
And the Koeins by
Defined.
Defines the antipode.
Group -like and primitive elements
An element of a Hopf algebra is called " group -like" when and. Then for the antipode.
An element is called " primitive " when. It follows that and.
An element is called " skew primitive " when and with group -like elements. It follows that and.