field , the k k -valued k k functions on a finite group form a Hopf algebra. Given a subgroup , there is an induced map of Hopf algebras B ⊂ G B\subset G , which is a k [ G ] → k [ B ] k[G]\to k[B] surjective homomorphism of commutative Hopf algebras. Similarly, for Hopf algebras of regular functions 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 sub semigroup.
However there is also a weaker notion of a quantum subgroup, and also a dual notion (e.g. via
coideal subalgebras). Definition
- k k bialgebra , a H H quotient bialgebra is a bialgebra equipped with an Q Q epimorphism of bialgebras . π : H → Q \pi: H\to Q
If both bialgebras are
Hopf algebras then the epimorphism will automatically preserve the antipode. Quotient bialgebras from bialgebra ideals
is an bialgebra ideal ideal in the sense of associative unital algebras which is also a coideal of coassociative coalgebras.
is a bialgebra ideal which is invariant under the Hopf ideal antipode map.
is a bialgebra and H H a bialgebra ideal then the quotient associative algebra I ⊂ H I\subset H has a natural structure of a bialgebra. Moreover, if H / I H/I is a Hopf algebra and H H is a Hopf ideal then the projection I ⊂ H I\subset H will be an epimorphism of Hopf algebras. H → H / I H\to H/I
Revised on September 23, 2010 03:25:28