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1、EXAMPLESOFFINITE-DIMENSIONALHOPFALGEBRASWITHTHEDUALCHEVALLEYPROPERTYNICOLASANDRUSKIEWITSCH,C′ESARGALINDOANDMONIQUEM′ULLER¨Abstract.WepresentnewHopfalgebraswiththedualChevalleypropertybydeterminingallsemisimpleHopfalgebrasMorita-equivalenttoagroupalgebraovera
2、?nitegroup,foralistofgroupssupportinganon-trivial?nite-dimensionalNicholsalgebra.1.Introduction1.1.AHopfalgebrahasthedualChevalleypropertyifthetensorproductoftwosimplecomodulesissemisimple,orequivalentlyifitscoradicalisa(cosemisimple)Hopfsubalgebra.TheseHopf
3、algebrasareinterestingbyvariousreasons,amongthemtheLiftingMethodfortheirclassi?cation[AS].Particularclassesarethepointed(thecoradicalisagroupalgebra)andcopointed(thecoradicalisthealgebraoffunctionsona?niteorreductivegroup)Hopfalgebras.Howeverfewexamplesoutof
4、theseclasseshasbeendiscussedintheliterature[CDMM,Mm].ThepurposeofthispaperistopresentexplicitexamplesofHopfalgebraswiththedualChevalleyproperty.1.2.Inthispapertheunderlying?eldkisalgebraicallyclosedofcharac-teristic0.Thenthecoradicalofa?nite-dimensionalHopfa
5、lgebrawiththedualChevalleypropertyisasemisimpleHopfalgebra.Letusconsidertheproblemofconstructing(andeventuallyclassifying)?nite-dimensionalHopfalgebrasAwhosecoradicalA0isisomorphictoa?xedsemisimpleHopfalgebraH.Forthis,weneedtoaddressthreeproblems:arXiv:1509.
6、01548v2[math.QA]14Oct2016H(a)To?nd(classify)theYetter-DrinfeldmodulesV∈HYDsuchthatthedimensionoftheNicholsalgebraB(V)is?nite.ThenA(V)=B(V)#Hisa?nite-dimensionalHopfalgebrawithA(V)0?H.(b)To?nd(classify)thedeformations,orliftings,ofA(V);thatis,HopfalgebrasAsuc
7、hthatgrA(thegradedHopfalgebrawithrespectthecoradical?ltration)isisomorphictoA(V).2010MathematicsSubjectClassi?cation.16T05.N.A.andM.M.werepartiallysupportedbyANPCyT-Foncyt,CONICETandSecyt(UNC).C.G.waspartiallysupportedbytheFAPAfundsfromvicerrectoriadeinvesti
8、-gacionesdelaUniversidaddelosAndes.PartofthisworkwasdoneduringvisitsofN.A.totheUniversidaddelosAndes.12N.ANDRUSKIEWITSCH,C.GALINDOANDM.MULLER¨(c)IfAisa?nite-dimensionalHopfalgebrawithA0?H,thenpr
