Bialgebra

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In mathematics, a bialgebra over a field K is a structure which is both a unital associative algebra and a coalgebra over K, such that the comultiplication and the counit are both unital algebra homomorphisms. Equivalently, one may require that the multiplication and the unit of the algebra both be coalgebra morphisms. The compatibility conditions can also be expressed by the following commutative diagrams:

The antipode:

Image:Bialgebra1.png

Multiplication and co-multiplication:

Image:Bialgebra2.png

Multiplication and counit:

Image:Bialgebra3.png

Multiplication and unit:

Image:Bialgebra4.png

Here <math>\nabla :B \otimes B \to B</math> is the algebra multiplication and <math>\eta: K \to B\,</math> is the unit of the algebra. <math>\Delta: B \to B \otimes B</math> is the comultiplication and <math>\epsilon: B \to K\,</math> is the counit. <math>\tau: B \otimes B \to B \otimes B</math> is the linear map defined by <math>\tau(x \otimes y) = y\otimes x</math> for all x and y in B.

In formulas, the bialgebra compatibility conditions look as follows (using the sumless Sweedler notation):

<math>(ab)_{(1)}\otimes (ab)_{(2)} = a_{(1)}b_{(1)} \otimes a_{(2)}b_{(2)}\, </math>

<math>1_{(1)}\otimes 1_{(2)} = 1 \otimes 1 \,</math>

<math>\varepsilon(ab)=\varepsilon(a)\varepsilon(b)\;</math>

<math>\varepsilon(1)=1.\;</math>

Here we wrote the algebra multiplication as simple juxtaposition, and 1 is the multiplicative identity.

Examples of bialgebras include the Hopf algebras and the Lie bialgebras, which are bialgebras with certain additional structure. Additional examples are given in the article on coalgebras.