Coalgebra

A coalgebra is a vector space, which has the dual to an algebra structure. That is, instead of a multiplication, which maps two elements to their product, there is a comultiplication that maps an element of a tensor, and instead of a neutral element, which allows the embedding of the body in the algebra, there is a mapping from the coalgebra in the base body, which is called Koeins.

Definition

A coalgebra over a field is a vector space with Vektorraumhomomorphismen, called comultiplication, coproduct, or diagonal, and called Koeins so that

A Koalgebrahomomorphismus between two coalgebras C and D is a homomorphism of vector with

Example

Be the canonical basis of. It is a coalgebra structure by means of

And

Define.

Is koassoziativ because

And is Koeins because

The elements of the second stage are tensors and therefore can be represented as matrices. The comultiplication is then

Duality

The multiplication of a ( unitary associative ) algebra is bilinear, and due to the universal property of the tensor product, it can be construed as a representation of after. The multiplication is associative if and only if the following diagram commutes.

An algebra has exactly then a neutral element, if there is a homomorphism of vector, so that the following diagram commutes:

In this case.

A coalgebra is an algebra in the dual vector spaces to the category. That is, instead of the multiplication there is a picture, so that the following dual diagram commutated:

And instead of a neutral element, there is a picture, so that the following dual diagram commutated:

Sweedlernotation

About the coproduct of an element is only known in general that it is in and, consequently, as

Can be represented. In the Sweedler notation, this is abbreviated by symbolically

Writes. In sum loose Sweedler notation is dispensed even on the sum symbol and writes

It is important to note that this notation is still referred to the sum. The symbols and are only meaningless for themselves and are not available for certain items from.

This notation allows the composition than other functions

To write.

In sum loose Sweedler notation is exactly then Koeins when

The coproduct is exactly then koassoziativ when

This element is symbolic as in Sweedler notation

And summenlos as

Written.

By reapply results in longer tensor that are written the same way. It must be the " indices " of the posterior elements increase if necessary:

By applying the tensor products that are adjusted accordingly, " indices " of the posterior elements shorten: