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S kd_t S kd. , /2. ,dl aLl natural , rrithout dim Z - dim X1 = 1 indecomposable with Then, by of root is system of Dynkin graphs says, as sum of roots, also trivial. numbers

J Mk ol be \, j . --r. 3)). M ,0), -32define and (sl s) is * u . O. , xI s t{k . 1 Lenma: + j€f o' by followed n (r = ,n* ,ä, ( Zi = t Si x')t - , "ru i€f q 1. 6 l(M,s*0). rt is easy to see, ttrat sl right 11g,sno). the diagram Mk "i i, jti ) isagainamorPhismin "i, then define sf S = ß = (ß1) by ß, = c, for and ßn : Zn and sl i+ k+ j, injection for [+I'= q = (si)r tr = (zr'jni)€t1rtr,s*a) = ! , "i "j L ( M, n)into from Let now k be a source. Then define s; I as follows, for ):lr*irj*x. e. by restriction, tig L( M, Q ), i+ k to be the map induced by also becomes a functor and sf, are additive from l(M,O) and sf, is left exact.

3) field and to Dynkin graphs. realisation (f,d) . ,Ia such that and Z be indecomposable modules -57- J dimZ= Then there is a sequence o = Zo c 21 c tion n of -.. t-1 there is {k1,. Proof: Lt, all for ... s kd_t S kd. , /2. ,dl aLl natural , rrithout dim Z - dim X1 = 1 indecomposable with Then, by of root is system of Dynkin graphs says, as sum of roots, also trivial. numbers

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