Depth two Hopf subalgebras of a semisimple Hopf algebra

Burciu, Sebastian; Kadison, Lars

doi:10.1016/j.jalgebra.2009.03.015

Cited by 6 publications

(3 citation statements)

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“…By means of general theory in one direction and Mackey theory in the other, depth two subgroup is shown to be precisely a normal subgroup [10]. (A similar statement is true for semisimple Hopf C -subalgebras [4].) In this paper we generalize this approach to depth two subgroup to a semisimple subalgebra pair, giving a condition in terms of inclusion matrix [7], which is the same as a induction-restriction table [1] up to a permutation change of basis.…”

Section: Introductionmentioning

confidence: 75%

“…There it is determined that a tower of three group algebras corresponding to the subgroup chain G ≥ H ≥ K is depth-3 if the normal closure K G (of K in G) is contained in H (and shown above in Theorem 1.1 to be a characterization of depth-3 tower of finite groups). In [10] it is shown that, with k = C and G a finite group, the group algebra A of G is depth two over subgroup algebra B of H if and only if H is a normal subgroup of G. This normality result for depth two subalgebras is extended to semisimple Hopf algebras over an algebraically closed field of characteristic zero in [4].…”

Section: Proposition 113 Let B Be a Hopf Subalgebra Of A Thenmentioning

confidence: 96%

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Subgroups of depth three

Burciu

Kadison

2010

Surveys in Differential Geometry

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A subalgebra pair of semisimple complex algebras B ⊆ A with inclusion matrix M is depth two iff 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. A tower of Hopf algebras A ⊇ B ⊇ C is shown to be depth-3 if C ⊂ core(B); and this is also a necessary condition if A, B and C are group algebras.

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Section: Introductionmentioning

confidence: 75%

Section: Proposition 113 Let B Be a Hopf Subalgebra Of A Thenmentioning

confidence: 96%

Subgroups of depth three

Burciu

Kadison

2010

Surveys in Differential Geometry

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“…Depth two is an algebraic notion for noncommutative ring extensions with a Galois theory associated to it [14,15]. If applied to a subalgebra pair of quantum algebras, depth two is a notion of normality that extends ordinary normality for subgroups and Hopf subalgebras [17,13,7]. A Hopf subalgebra K is normal in a finite dimensional Hopf algebra H if and only if H is a depth two ring extension of K: see [3] for this theorem and how it generalizes to faithfully flat, finitely generated projective Hopf algebra extensions over a commutative base ring as well as one-sided versions of normality, depth two and Hopf-Galois extension.…”

Section: Introductionmentioning

confidence: 99%

On subgroup depth

Burciu,

Kadison,

Kuelshammer

2009

Preprint

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We define a notion of depth for an inclusion of multimatrix algebras B ⊆ A based on a comparison of powers of the inductionrestriction table M (and its transpose matrix). This notion of depth coincides with the depth from [16]. In particular depth 2 extensions coincide with normal extensions as introduced by Rieffel in [20]. 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 Sn < Sn+1 is 2n − 1. An appendix by S. Danz and B. Külshammer determines the subgroup depth of alternating groups An < An+1 and of some other group extensions.Remark A.6. Note that the ring extensions CS 1 ⊂ CS 2 and CA 2 ⊂ CA 3 are clearly of depth 2, since S 1 ✂ S 2 and A 2 ✂ A 3 .

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