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