Knots and algebras H.R.Morton and P.Traczyk

《Knots and algebras H.R.Morton and P.Traczyk》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、KnotsandalgebrasH.R.MortonandP.Traczyk*Abstract.Startingfromtheexistenceofthe2-variablepolynomialPfororientedlinkswedevelopthelinearskeintheoryapproachtogiveageometricrealisationoftheHeckealgebras.AcarefuldenitionoftheringinwhichPtakesvaluesmakesiteasierthanu

2、sualtostudytherepresentationsofpiecesofaknotdiagramintheHeckealgebrasandinspecialisationsofthemsuchasthegroupalgebrasZ[S]:nAnanalogousmethodisusedtoconstructalgebrasbasedonKauman'spolyno-mial.Hereasimilarlycarefulchoiceofring,coupledwiththeuseoftheDubrovnikvar

3、iantofthepolynomialallowsanaturalspecialisationtoBrauer'salgebras.Introduction.The2-variablepolynomialP(K)ofanorientedlinkKwasdevelopedfromtwodierentstartingpoints.OneapproachwasthroughtheHeckealgebrasHn[J1,J2,O],whiletheothercombinatorialapproachwasthroughkno

4、tdiagramsandeventuallylinearskeintheory,[FYHLMO,PT].Weuseherethelinearskeintheoryapproachbasedontangles(piecesofaknotdiagram)togiveageometricrealisationoftheHeckealgebras,assumingtheexistenceofP(K):Oneimportantfeatureintheconstructionaspresentedhereisthedeniti

5、onoftheringinwhichP(K)takesvalues,tobeasubringofthemoreusualLaurentpolynomialring.Thisallowsvariousspecialisationsofthealgebras,whichresultfromspecialising;tobereadilyconstructed.Weusesimilarmethods,startingwithKauman'spolynomialinitsDubrovnikform[K],againke

6、epingaclosewatchontheringused,toconstructalgebraswhosespecialisationscanbeidentiedwithBrauer'salgebras[B]inaverynaturalway.SuchalgebrashavebeenstudiedinmoredetailbyBirmanandWenzl[BW],anditwasfromthemthatwegottohearofBrauer'salgebras.Whiletheiralgebrasareessent

7、iallyisomorphictotheonesconstructedhere,ourchoiceofringandtheuseoftheDubrovnikvariantofthepolynomialallowustomaketheirconjecturedconnectionwithBrauer'salgebrasverydirectlybothalgebraicallyandgeometrically.Forthisreason,andalsobecauseoftheformofrecentlydiscovere

8、dconnectionsofbothpolynomialswithinvariantsderivedfromLiealgebras[T],wefeelthattheDubrovnikversion,andtheversionofPwhichweuse,bothhaveaparticularlyappropriatechoiceofvariabl