nLab quotient bialgebra

Geometric motivation

Given a field $k$, the $k$-valued functions on a finite group form a Hopf algebra. Given a subgroup $B\subset G$, there is an induced map of Hopf algebras $k[G]\to k[B]$, which is a surjective homomorphism of commutative Hopf algebras. 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 $k$-bialgebra $H$, a quotient bialgebra is a bialgebra $Q$ equipped with an epimorphism of bialgebras $\pi: H\to Q$.

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