nLab
bialgebra

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 MM is a monoid in the category of comonoids in MM — or equivalently, a comonoid in the category of monoids in MM. So, a bialgebra is a bimonoid in VectVect.

Properties

Relation to sesquialgebras

Bialgebras are precisely those sesquialgebras AA whose product AAA \otimes A-AA-bimodule is induced from an algebra homomorphism AAAA \to A \otimes A and whose unit kk-AA bimodule is induced from an algebra homomorphism AkA \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 algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

Examples

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

References

Tannaka duality for bialgebras

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

Revised on July 11, 2013 11:49:00 by Urs Schreiber (82.113.106.129)