Bialgebra

Prejudice to the special areas

• Mathematics Abstract Algebra
• Linear Algebra
• Commutative Algebra

Is a special case of

• Algebra
• Coalgebra

Includes as special cases

• Hopf algebra

A Bialgebra has both the structure of a unitary associative algebra and the corresponding dual structure of a coalgebra. The most important special case of Hopf algebras are Bialgebren, which include the quantum groups.

Definition

Be a body and both unitary associative algebra over and coalgebra over. This denotes the multiplication, the one ( embedding of the body in the algebra ) that comultiplication and Koeins.

Ie Bialgebra over if the following equivalent compatibility conditions are satisfied.

• The comultiplication and the Koeins are Algebrahomomorphismen.
• The multiplication and the One are Koalgebrahomomorphismen.
• The following diagrams commute

This is the " flip" picture, so the canonical isomorphism of the tensor and applied.

The Bialgebren together with the pictures that are both algebra and Koalgebrahomomorphismen, a category.

Generalization

Algebras and coalgebras can be viewed in any monoidal categories. However, for compatibility conditions, it is necessary that the tensor product of a ( co ) algebra in a natural way is a ( co ) algebra again, this implies the existence of a knot style.