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.

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.

Last revised on September 23, 2010 at 03:25:28.
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