Bialgebra

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 diagrams".

As reflected in the symmetry of the diagrams, the definition of bialgebra is self-dual, so if one can define a dual of "B" (which is always possible if "B" is finite-dimensional), then it is automatically a bialgebra.

Diagrams

The compatibility conditions can be expressed by the following commutative diagrams, 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::

Here $ablacolon\; B\; otimes\; B\; o\; B$ is the algebra multiplication and $etacolon\; K\; o\; B,$ is the unit of the algebra. $Deltacolon\; B\; o\; B\; otimes\; B$ is the comultiplication and $varepsiloncolon\; B\; o\; K,$ is the counit. $aucolon\; B\; otimes\; B\; o\; B\; otimes\; B$ is the linear map defined by $au(x\; otimes\; y)\; =\; yotimes\; x$ 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::$varepsilon(ab)=varepsilon(a)varepsilon(b);$

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

Unit and counit::$varepsilon(1)=1.;$

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

=See also=
*Quasi-bialgebra
*Hopf algebra