arXiv subjects
Publications and source records attributed to Sebastian Burciu.
In the category of semisimple Hopf algebras the Hopf kernels introduced by Andruskiewitsch and Devoto in \cite{AD} coincide with kernels of representation as introduced in \cite{Bker}. Some new results concerning the normality of kernels are also presented. It is proven that the property for Hopf algebras to have all kernels normal Hopf subalgebras is a self dual property.
A subalgebra pair of semisimple complex algebras B < A with inclusion matrix M is depth two if MM^t M < nM for some positive integer n and all corresponding entries. If A and B are the group algebras of finite group-subgroup pair H < G, the induction-restriction table equals M and S = MM^t satisfies S^2 < nS iff the subgroup H is depth three in G; similarly depth n > 3 by successive right multiplications of this inequality with alternately M and M^t. We show that a Frobenius complement in a Frobenius group is a nontrivial class of examples of depth three subgroups. Depth-3 towers of Hopf algebras are also considered: a tower of Hopf algebras A > B > C is shown to be depth-3 if C < core(B).
We define a notion of depth for an inclusion of multimatrix algebras B < A based on a comparison of powers of the induction-restriction table M (and its transpose matrix). This notion of depth coincides with the depth from [Kadison, 2008]. In particular depth 2 extensions coincides with normal extensions as introduced by Rieffel in 1979. For a group extension H < G a necessary depth n condition is given in terms of the core of H in G. We prove that the subgroup depth of symmetric groups S_n < S_{n+1} is 2n-1. An appendix by S. Danz and B. Kuelshammer determines the subgroup depth of alternating groups A_n < A_{n+1} as well as dihedral groups.
For any normal commutative Hopf subalgebra $K=k^G$ of a semisimple Hopf algebra we describe the ring inside $kG$ obtained by the restriction of $H$-modules. If $G=\Z_p$ this ring determines a fusion ring and we give a complete description for it. The case $G=\Z_{p^n}$ and some other applications are presented.
Let H be a finite dimensional semisimple Hopf algebra over an algebraically closed field of characteristic zero. In this note we give a short proof of the fact that a Hopf subalgebra of H is a depth two subalgebra if and only if it is normal Hopf subalgebra.