Semisimple Reflection Hopf Algebras of Dimension Sixteen

Ferraro, Luigi; Kirkman, Ellen; Moore, W. Frank; Won, Robert

doi:10.1007/s10468-021-10038-w

Cited by 2 publications

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“…In [17] the case of a Hopf algebra of the form H = kG • , the dual of a group algebra (or equivalently, a group coaction) was considered, and some dual reflection groups were constructed. In [10] the sixteen non-trivial Hopf algebras of dimension sixteen classified by Kashina [12] were considered, and the methods used in this paper were used to determine which are reflection Hopf algebras for AS regular algebras of dimension 2 and 3.…”

Section: Introductionmentioning

confidence: 99%

Three infinite families of reflection Hopf algebras

Ferraro

Kirkman

Moore

et al. 2020

Journal of Pure and Applied Algebra

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Let H be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra A, homogeneously, inner-faithfully, preserving the grading on A, and so that A is an H-module algebra. When the fixed subring A H is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that H is a reflection Hopf algebra for A. We show that each of the semisimple Hopf algebras H 2n 2 of Pansera, and A 4m and B 4m of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3.2010 Mathematics Subject Classification. Primary 16T05, 16E65, 16G10.The Hopf algebras A 4m and B 4m can be viewed as deformations of kQ 4m (see [2]), and are extensions of the form:The examples of reflection Hopf algebras that we have computed indicate that there are an abundance of examples. The properties that characterize such a pair (A, H) are not clear, and invite further investigation. One obvious question is:Question 0.1. When is a bicrossed product H = K# τ σ H a reflection Hopf algebra for some AS regular algebra A?The method that is used in this paper is as follows. First, we compute the Grothendieck ring of finite-dimensional H-modules for each Hopf algebra H. The results are summarized in the following table.

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Section: Introductionmentioning

confidence: 99%