On pointed Hopf algebras over dihedral groups

Abstract

Let k be an algebraically closed field of characteristic 0 and let D_m be the dihedral group of order 2m with m= 4t, with t bigger than 2. We classify all finite-dimensional Nichols algebras over D_m and all finite-dimensional pointed Hopf algebras whose group of group-likes is D_m, by means of the lifting method. Our main result gives an infinite family of non-abelian groups where the classification of finite-dimensional pointed Hopf algebras is completed. Moreover, it provides for each dihedral group infinitely many non-trivial new examples.

23 pages, 1 figure and several tables

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Cited by in corpus (11)

• Finite dimensional Hopf algebras over Kac-Paljutkin algebra $H_8$
• On Hopf Algebras over quantum subgroups
• On finite-dimensional copointed Hopf algebras over dihedral groups
• Deformation by cocycles of pointed Hopf algebras over non-abelian groups
• Finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}$ of Kashina
• On some classification of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}^{\ast }$ of Kashina
• On rigidity of Nichols algebras
• Simple modules of small quantum groups at dihedral groups
• Twist equivalence for Nichols algebras over Coxeter groups
• On Hopf algebras whose coradical is a cocentral abelian cleft extension
• Ext algebra of Nichols algebras of type $A_2$