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1、˖ڍመڙጲhttp://www.paper.edu.cn对称线性Gr-范畴中的李代数黄华林1，杨毓萍21华侨大学数学科学学院，泉州 3620212西南大学数学与统计学院，重庆 400715摘要：本文中，我们主要关注那些具有非平凡结合性质的对称线性Gr-范畴中的李代数。这些李代数乃是具有一般结合线性GR范畴中的color李代数的自然对应。我们给出在对称线性GR范畴中李代数的约化形式，并且对一般李代数、李超代数和color李代数的统一形式的PBW定理。关键词：李代数，对称范畴，辫子，上循环，GR范畴中图分类号：O153,O15
2、4LIEALGEBRASINSYMMETRICLINEARGR-CATEGORIESHua-LinHuang1,YupingYang21SchoolofMathematicalSciences,HuaqiaoUniversity,Quanzhou3620212SchoolofMathematicsandStatistics,SouthwestUniversity,Chongqing400715Abstract:Inthispaper,westudyLiealgebrasinsymmetriclinearGr-categories
3、withfocusonthosewithnontrivialassociativityconstraints.SuchLiealgebrasarenaturalcounterpartsofthewellknownLiecoloralgebraswhichliveinlinearGr-categorieswithassociativityisomorphismsbeingidentity.WegivethereducedformoftheLiealgebrasinsymmetriclinearGR-categoryandprove
4、thePBWtheoremwhichisaunifiedformofthatforordinaryLiealgebras,LiesuperalgebrasandLiecoloralgebras.Keywords:Liealgebra,symmetriccategory,braiding,cocycle,Gr-categoryFoundations:SRFDP20130131110001AuthorIntroduction:Hua-LinHuang（1975-），male，professor，majorresearchdirecti
5、on：algebra,Email:hualin.huang@hqu.edu.cn.Correspondenceauthor：YupingYang（1987-），male，assistantprofessor，majorresearchdi-rection：algebra,Email:yupingyang@swu.edu.cn.-1-˖ڍመڙጲhttp://www.paper.edu.cn0IntroductionBraidedtensorcategoriesareinterestedmainlybecauseofthede
6、epconnectionwithquantumgroupsandphysics.Ontheonehand,alargeclassesofbraidedtensorcategoriesisrepresentationcategoriesofHopfalgebrasandhencethestructureofsuchcategoriescanbeinvestigatedbystudyingthecorrespondingHopfalgebras;ontheanotherhand,thestructureofBraidedtensor
7、categoriesandthealgebrasinBraidedtensorcategoriesprovidemuchinformationoftherepresentationsandstructuretheoriesofHopfalgebras.ItiswellknownthatthebraidingofBraidedtensorcategoriesprovidesystematicsolutionsofYang-Baxterequation,andthisisoneofthemainreasonswhyBraidedte
8、nsorcategoriesandquantumgroupsarewidelyappliedinphysicsandintegrablesystems.Symmetriccategoryisabraidedtensorcategorywithsymmetricb
