We use the idea of generic extensions to investigate the correspondence between the isomorphism classes of nilpotent representations of a cyclic quiver and the orbits in the corresponding representation varieties. We endow the set $\cal M$ of such isoclass

2BANGMINGDENGANDJIEDU

so-calledcondensedwordswhosede nitionisratherlongandcomplicated.Inthispaper,weshalldescribemonomialbasesforaquantuma neslninamoregeneralandsatisfactoryway.Weshallprovethefollowingmonomialbasistheorem(see§8).

n).Thenthereisapartition =∪π∈Πs πsuchthat,forTheorem1.1.LetU=Uv(sl

anysubsets ± with| ±∩ π|=1foreveryπ∈Πs,theset

{FyKaEw|y∈ ,w∈ +,a∈ZI}

formsabasisforU.

Theproofofthetheoremusestheideaofgenericextensionsofnilpotentrepresentationsofacyclicquiverwithnvertices.We rstgeneralizearecentworkbyReineke[10]toobtainamonoidstructureonthesetMofisoclassesofnilpotentrepresentationsindexedbyΠ,thesetofn-tuplesofpartitions.SimplemoduleswillgenerateasubmonoidMc.Thus,everywordw∈ de nesauniqueisoclassinMcwhichisindexedby (w).Weshallprovethatthesetofall (w)coincideswiththesubsetΠsofseparatedmultipartitionsde nedin[13,4.1].Thus,the bresof yieldapartitionof .Ourargumentwillthendependheavilyonthestructureofnilpotentrepresentations.

Weorganisethepaperasfollows.Westartwithinvestigatingseveralusefulpropertiesofnilpotentrepresentationsin§2.ThenwemoveonlookingatgenericextensionsofnilpotentrepresentationsandconstructingthemonoidMin§3.In§4,wediscusstherelationbe-tweenthesubmonoidMcandseparatedmultipartitions.Somealgorithmsareintroducedtocalculatethemultipartition (w)andthewordsinthe bresof .Thenotionofdistin-guishedwordsisde nedintermsofacertainmodulestructure.Howeveracombinatorialcriterionexists.Thesewillbediscussedin§5.Therearetwopartialorderrelationsonnilpotentrepresentationsde nedgeometricallybytheinclusiverelationontheclosuresoforbitsandalgebraicallybymoduleextensions.Weshallprovein§6thatthetworelationscoincide.Moreover,wedescribeaposetidealgeneratedbyanelementofMcintermsoftheexistenceofcertaincompositionseries.ThisisTheorem6.3.Someapplicationsaregiveninthelastthreesections.In§7,wegeneralizesomeresultsofRingelgivenin[13],andin§8,weproveTheorem1.1.Finally,inthelastsection,weconstructaPBWtypebasisforU+andspeculatesomerelationsbetweenthevariousbasesincludingthecanonicalbasis.

Inaforthcomingpaper,weexpecttoproveasimilarresultforthequantumgroupsofsimply-laced nitetype.¯Somenotation.Throughoutthispaper,kdenotesa eld.Weshallassumethatk=kisalgebraicallyclosedinsectionthree.AllmodulesMare nitedimensionaloverk.Wedenotebyrad(M)theradicalofM,i.e.,theintersectionofallmaximalsubmodulesofM,andbytop(M):=M/rad(M)thetopofM.

Thefollowinglemmaisaresultofapull-backorpush-outdiagram.

Lemma1.2.Let0→X→M→Y→0beashortexactsequenceofmodules.Thenanyexactsequence0→Y′→Y→Y′′→0givesrisetoacommutativediagramwithexactrowsandcolumns