nLab
bialgebra

Idea

A bialgebra (or bigebra) is both an algebra and a coalgebra, where the operations of either one are homomorphisms for the other. A bialgebra structure on an associative algebra is precisely such as to make its category of modules into a monoidal category equipped with a fiber functor.

A bialgebra is one of the ingredients in the concept of Hopf algebra.

Definition

A bialgebra is a monoid in the category of coalgebras. Equivalently, it is a comonoid in the category of algebras. Equivalently, it is a monoid in the category of comonoids in Vect — or equivalently, a comonoid in the category of monoids in Vect.

More generally, a bimonoid in a monoidal category $M$ is a monoid in the category of comonoids in $M$ — or equivalently, a comonoid in the category of monoids in $M$. So, a bialgebra is a bimonoid in $\mathrm{Vect}$.

Properties

Relation to sesquialgebras

Bialgebras are precisely those sesquialgebras $A$ whose product $A\otimes A$-$A$-bimodule is induced from an algebra homomorphism $A\to A\otimes A$ and whose unit $k$-$A$ bimodule is induced from an algebra homomorphism $A\to k$.

Tannaka duality and categories of modules

The structure of a bialgebra on an associative algebra equips its category of modules with the structure of a monoidal category and a monoidal fiber functor. In fact that construction is an equivalence. This is the statement of Tannaka duality for bialgebras. For instance (Bakke)

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	${\mathrm{Mod}}_A$
$R$-algebra	${\mathrm{Mod}}_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	${\mathrm{Mod}}_A$
$R$-2-algebra	${\mathrm{Mod}}_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category	3-category of module 2-categories
$A$	${\mathrm{Mod}}_A$
$R$-3-algebra	${\mathrm{Mod}}_R$-4-module

Examples

Notions of bialgebra with further structure notably include Hopf algebras and their variants.

Related concepts

• Gerstenhaber-Schack cohomology, bialgebra cocycle, weak bialgebra, bialgebroid

• Hopf algebra

• quasitriangular bialgebra

• sesquialgebra

• algebra, module

• bialgebra, 2-module

• trialgebra, 3-module

References

Tannaka duality for bialgebras

• Tørris Koløen Bakke, Hopf algebras and monoidal categories (2007) (pdf)