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By V. B Dlab

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Extra resources for An Introduction to Diagrammatical Methods in Representation Theory

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Srrn (Uy "1 therefore we can apply first ,i) if) =t Q) and an s1 0 and sirnilar be used throughout LetX6 the ... r1*r I. Then the following i) ... €Stl. 3proposition: a) ... for simple module in "t_2 L ( M, sa*' get finalty l/, s1 ... t), rjl "o*, statements: - - Jö We prove b): Proof: - ii) iii) = s: t I ... -. "ri ..... + ii) iii1 r s; t" ... tl*, "i*, imPlies I. = o implies that there is a 11 t1 n I ... t" I = sl I = o. with sl*, ... s" x = y +9 uut sl ti*, Hence by iii)r r ii): they are i! a , 11 t annihilated by All indecomposable by vre get modules, hetrce all ...

T st_1 st X is projective. iii) ... e ) . rn st ... srrn (Uy "1 therefore we can apply first ,i) if) =t Q) and an s1 0 and sirnilar be used throughout LetX6 the ... r1*r I. Then the following i) ... €Stl. 3proposition: a) ... for simple module in "t_2 L ( M, sa*' get finalty l/, s1 ... t), rjl "o*, statements: - - Jö We prove b): Proof: - ii) iii) = s: t I ... -. "ri ..... + ii) iii1 r s; t" ... tl*, "i*, imPlies I. = o implies that there is a 11 t1 n I ... t" I = sl I = o. with sl*, ... s" x = y +9 uut sl ti*, Hence by iii)r r ii): they are i!

S, rr S . ,! , ! dim such rhat Z = O. Then S. " S, ... - . x. ,r't X >O and si ... u .. e, because s. y ^1 - > O. Thus Z ' = s-;r . . s-]1 z + o . 1) yields an exact sesuence -54- O+4,- sl- ... 112 Z_ +1. A. celfand all graphs i. j 9. M. d) closed done by V. arly where K. N. by for quivers modulation Rj-ngel in t251. Bernstein, where d.. M. in for [15]. e. s given $ 2 see = dli by an [11] - Chapter fII Graphs of An abelian tion category series A A and is has only objects, (il,o), if for then of all is for Artin K', K' finite to the L(M'o) denotes the category type, generated exact to the is free x assoy.

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