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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 these structures are compatible.

Compatibility means that the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, that the multiplication and the unit of the algebra both be coalgebra morphisms: these statements are equivalent in that they are expressed by ''the same diagram''.


Diagrams

The compatibility conditions can be expressed by the following Commutative Diagram s, which can be read either as "comultiplication is a map of algebras" or "multiplication is a map of coalgebras" (similarly for the others):

Multiplication and comultiplication:
:

Multiplication and counit:
:

Comultiplication and unit:
:

Unit and counit (the antipode):
:

Here
abla\colon B \otimes B o B is the algebra multiplication and \eta\colon K o B\, is the unit of the algebra. \Delta\colon B o B \otimes B is the comultiplication and \epsilon\colon B o K\, is the counit. au\colon B \otimes B o B \otimes B is the Linear Map defined by au(x \otimes y) = y\otimes x for all ''x'' and ''y'' in ''B''.


Formulas

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

Multiplication and comultiplication:
:(ab)_{(1)}\otimes (ab)_{(2)} = a_{(1)}b_{(1)} \otimes a_{(2)}b_{(2)}\,

Multiplication and counit:
: arepsilon(ab)= arepsilon(a) arepsilon(b)\;

Comultiplication and unit:
:1_{(1)}\otimes 1_{(2)} = 1 \otimes 1 \,

Unit and counit:
: arepsilon(1)=1.\;

Here we suppressed the algebra notation: we wrote the algebra multiplication
abla as simple juxtaposition, and the unit \eta via the multiplicative identity 1.


EXAMPLES

Examples of bialgebras include the Hopf Algebra s and the Lie Bialgebra s, which are bialgebras with certain additional structure. Additional examples are given in the article on Coalgebra s.

=See Also=