▁e ▁is ▁part ▁of ▁an ▁sl 2 ▁triple ▁( e , h , f ). ▁The ▁eig ens pace ▁decomposition ▁of ▁ad ( h ) ▁indu ces ▁a ▁- grad ing ▁on ▁g : ▁ ▁Define ▁a ▁character ▁ ▁( i . e . ▁a ▁hom omorphism ▁from ▁g ▁to ▁the ▁trivial ▁ 1 - dimensional ▁Lie ▁algebra ) ▁by ▁the ▁rule ▁, ▁where ▁ ▁denotes ▁the ▁K illing ▁form . ▁This ▁indu ces ▁a ▁non - de generate ▁anti - sym metric ▁bil ine ar ▁form ▁on ▁the ▁- 1 ▁grad ed ▁piece ▁by ▁the ▁rule : ▁ ▁After ▁choosing ▁any ▁L agr ang ian ▁sub space ▁, ▁we ▁may ▁define ▁the ▁following ▁nil pot ent ▁sub algebra ▁which ▁acts ▁on ▁the ▁universal ▁en velop ing ▁algebra ▁by ▁the ▁ad joint ▁action . ▁ ▁The ▁left ▁ideal ▁ ▁of ▁the ▁universal ▁en velop ing ▁algebra ▁ ▁generated ▁by ▁ ▁is ▁invariant ▁under ▁this ▁action . ▁It ▁follows ▁from ▁a ▁short ▁calculation ▁that ▁the ▁invari ants ▁in ▁ ▁under ▁ad ▁inherit ▁the ▁associ ative ▁algebra ▁structure ▁from ▁. ▁The ▁invariant ▁sub space ▁ ▁is ▁called ▁the ▁finite ▁W - algebra ▁constructed ▁from ▁( g , e ) ▁and ▁is ▁usually ▁denoted ▁. ▁ ▁References ▁ ▁Category : Con form al ▁field ▁theory ▁Category : In te gra ble ▁systems ▁Category : Re presentation ▁theory <0x0A> </s> ▁Four th ▁World ▁refers ▁to ▁a ▁sub - pop ulation ▁subject ed ▁to ▁social ▁ex clusion ▁in ▁global ▁society , ▁or ▁stat eless ▁and ▁not ably ▁imp