Semisimple Hopf algebras and their depth two Hopf subalgebras

Details Semisimple Hopf algebras and their depth two Hopf subalgebras Semisimple Hopf algebras and their depth two Hopf subalgebras

2008-07-03

arxiv.org/abs/0807.0599v1

Mathematics

Quantum Algebra

15 pages

Scientific paper

We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field $k$ is algebraically closed of characteristic zero). This means on the one hand that a Hopf subalgebra is normal when inducing (then restricting) modules several times as opposed to one time creates no new simple constituents. This point of view was taken in the paper http://arxiv.org/abs/math/0409346 which established a normality result in case H and K are finite group algebras. On the other hand this means that K is normal in H when H | K is a Galois extension with respect to action of generalized bialgebras such as bialgebroids, weak Hopf algebras or Hopf algebroids. The generalized Galois picture of depth two is the point of view we take here: after showing the centralizer R is separable algebra via Hopf invariant theory, we compute that the depth two semisimple Hopf algebra pair H | K is free Frobenius extension with Markov trace satisfying all hypotheses considered in Kadison and Nikshych, Frobenius Extensions and Weak Hopf Algebras, J.Algebra 244 (2001). By the main theorem in that paper it is then a Galois extension with action of semisimple weak Hopf algebra (also regular and possessing Haar integral). Then the Galois canonical isomorphism (via coring theory) restricted to integral induces algebra homomorphism from Hopf algebra into weak Hopf algebra with kernel HK^+ = K^+H.

Affiliated with

Also associated with

No associations

Rating

Semisimple Hopf algebras and their depth two Hopf subalgebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Semisimple Hopf algebras and their depth two Hopf subalgebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Semisimple Hopf algebras and their depth two Hopf subalgebras will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-116085