EXAMPLES OF FINITE-DIMENSIONAL HOPF ALGEBRASWITH THE DUAL CHEVALLEY PROPERTY有限维Hopf代数地例子 《房地产与双.pdf

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1、EXAMPLESOFFINITE-DIMENSIONALHOPFALGEBRASWITHTHEDUALCHEVALLEYPROPERTYNICOLASANDRUSKIEWITSCH,C´ESARGALINDOANDMONIQUEM´ULLER¨Abstract.WepresentnewHopfalgebraswiththedualChevalleypropertybydeterminingallsemisimpleHopfalgebrasMorita-equivalenttoagroupalgebraovera

2、finitegroup,foralistofgroupssupportinganon-trivialfinite-dimensionalNicholsalgebra.1.Introduction1.1.AHopfalgebrahasthedualChevalleypropertyifthetensorproductoftwosimplecomodulesissemisimple,orequivalentlyifitscoradicalisa(cosemisimple)Hopfsubalgebra.TheseHopf

3、algebrasareinterestingbyvariousreasons,amongthemtheLiftingMethodfortheirclassification[AS].Particularclassesarethepointed(thecoradicalisagroupalgebra)andcopointed(thecoradicalisthealgebraoffunctionsonafiniteorreductivegroup)Hopfalgebras.Howeverfewexamplesoutof

4、theseclasseshasbeendiscussedintheliterature[CDMM,Mm].ThepurposeofthispaperistopresentexplicitexamplesofHopfalgebraswiththedualChevalleyproperty.1.2.Inthispapertheunderlyingfieldkisalgebraicallyclosedofcharac-teristic0.Thenthecoradicalofafinite-dimensionalHopfa

5、lgebrawiththedualChevalleypropertyisasemisimpleHopfalgebra.Letusconsidertheproblemofconstructing(andeventuallyclassifying)finite-dimensionalHopfalgebrasAwhosecoradicalA0isisomorphictoafixedsemisimpleHopfalgebraH.Forthis,weneedtoaddressthreeproblems:arXiv:1509.

6、01548v2[math.QA]14Oct2016H(a)Tofind(classify)theYetter-DrinfeldmodulesV∈HYDsuchthatthedimensionoftheNicholsalgebraB(V)isfinite.ThenA(V)=B(V)#Hisafinite-dimensionalHopfalgebrawithA(V)0≃H.(b)Tofind(classify)thedeformations,orliftings,ofA(V);thatis,HopfalgebrasAsuc

7、hthatgrA(thegradedHopfalgebrawithrespectthecoradicalfiltration)isisomorphictoA(V).2010MathematicsSubjectClassification.16T05.N.A.andM.M.werepartiallysupportedbyANPCyT-Foncyt,CONICETandSecyt(UNC).C.G.waspartiallysupportedbytheFAPAfundsfromvicerrectoriadeinvesti

8、-gacionesdelaUniversidaddelosAndes.PartofthisworkwasdoneduringvisitsofN.A.totheUniversidaddelosAndes.12N.ANDRUSKIEWITSCH,C.GALINDOANDM.MULLER¨(c)IfAisafinite-dimensionalHopfalgebrawithA0≃H,thenpr