nLab
quotient bialgebra

Quotient bialgebras

Geometric motivation

Given a field kk, the kk-valued functions on a finite group form a Hopf algebra. Given a subgroup BGB\subset G, there is an induced map of Hopf algebras k[G]k[B]k[G]\to k[B], which is a surjective homomorphism of commutative Hopf algebra?s. Similarly, for Hopf algebras of regular function?s on an algebraic group over a field.

The generalization to noncommutative Hopf algebras hence may be viewed as describing the notion of a quantum subgroup?, or in the bialgebra version of a quantum subsemigroup.

However there is also a weaker notion of a quantum subgroup, and also a dual notion (e.g. via coideal subalgebras).

Definition

Given a kk-bialgebra HH, a quotient bialgebra is a bialgebra QQ equipped with an epimorphism of bialgebras π:HQ\pi: H\to Q.

If both bialgebras are Hopf algebras then the epimorphism will automatically preserve the antipode.

Quotient bialgebras from bialgebra ideals

A bialgebra ideal? is an ideal in the sense of associative unital algebras which is also a coideal of coassociative coalgebras.

A Hopf ideal is a bialgebra ideal which is invariant under the antipode map.

If HH is a bialgebra and IHI\subset H a bialgebra ideal then the quotient associative algebra H/IH/I has a natural structure of a bialgebra. Moreover, if HH is a Hopf algebra and IHI\subset H is a Hopf ideal then the projection HH/IH\to H/I will be an epimorphism of Hopf algebras.

Revised on September 23, 2010 03:25:28 by Toby Bartels (98.19.51.164)