Lectures on Representations of Quivers by William Crawley-Boevey ...

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¨ u © ¢ 0 0 v¢ which corresponds to U¥ V. Finally Hom( ¢ so ,U) gives an exact sequence f 1 0 ¡ Hom(V,U) ¡ Hom(X,U) ¡ Hom(U,U) ¡ Ext (V,U), dim Hom(V,U) - dim Hom(X,U) + dim Hom(U,U) - dim Im(f) = 0, but f(1 )= ¢ U CONSEQUENCES. £ 0, so dim Hom(X,U) £ dim Hom(U¥ V,U), and hence X ¦ U¥ V. (1) If O is an orbit Rep(¤ in ) of maximal dimension, X=U¥ and V, then X 1 Ext (V,U)=0. A PROOF. If there is non-split extension ¡ 0 ¡ U ¡ E ¡ V 0 then O O \O , so X E E dim O < dim O . X E (2) If O is closed then X is semisimple. X REMARKS. (1) Suppose Q has no oriented cycles. Let z¡ Rep(¤ ) be the element with all matrices z =0. We can easily show that z is in the closure of every orbit, ¢ and it follows that there are no non-constant polynomial invariants. Moreover, an orbit O is closed X is semisimple, for the only semisimple X module of dimension ¤ is R(z), and {z} is clearly a closed orbit. (2) If Q is allowed to have oriented cycles, x,x¥ ¡ Rep(¤ ) and R(x) and R(x¥ ) are non-isomorphic semisimple modules, then there is a polynomial invariant § (of the form f ) with § (x)£ § (y). In case Q has only one vertex and ai char k=0 this is proved in §12.6 of M.Artin, On Azumaya algebras and finite dimensional representations of rings, J.Algebra 11 (1969), 532-563, but it seems to be true in general. It follows that O is closed X is X semisimple. 14
§4. Dynkin and Euclidean diagrams In this section we give the classification of graphs into Dynkin, Euclidean, and ’wild’ graphs, and in the first two cases we study the corresponding root system. DEFINITIONS. Let be finite graph with vertices {1,..,n}. We allows loops and multiple edges, so that is given by any set of natural numbers n = n = the number of edges between i and j. ij ji i¤ n 2 i=1 i j ij i j Let q(¤ ) = ¤ ¤ - ¤ n ¤¢¤ n Let (-,-) be the symmetric bilinear form on ¦ with § ¨© (i£ -n ij ( , ) = i j 2 - 2n ii j) (i=j) where i th is the i coordinate vector. Note that knowledge of any one of , q or (-,-) determines the others, 1 since q(¤ ) = (¤ ,¤ ) and (¤ ,¥ ) = q(¤ +¥ )-q(¤ )-q(¥ ). 2 If Q is a quiver and is its underlying graph, then (-,-) and q are the same as before. The bilinear form , however, depends on the orientation of Q. n We say q is positive definite if q(¤ )>0 for all 0£ ¤ ¡ ¦ . n We say q is positive semi-definite if q(¤ )£ 0 for all ¤ ¡ ¦ . n The radical of q is rad(q) = {¤ ¡ ¦ ¢ (¤ ,-) = 0}. n n We have a partial ordering on ¦ given by ¤¦¤ ¥ if ¥ -¤ ¡ £ . n We say that ¤ ¡ ¦ is sincere if each component is non-zero. LEMMA. If is connected and ¥¦£ 0 is a non-zero radical vector, then ¥ is n sincere and q is positive semi-definite. For ¤ ¡ ¦ we have q(¤ )=0 ¢¤ ¡¢¡ ¥ ¢¤ ¡ rad(q). 15
Page 1 and 2: Lectures on Repres
Page 3 and 4: §1. Path algebras Once and for all
Page 5 and 6: PROPERTIES OF PATH ALGEBRAS. These
Page 7 and 8: THE STANDARD RESOLUTION. ¡ ¥ ¡
Page 9 and 10: §2. Bricks In this section we cons
Page 11 and 12: §3. The variety of representations
Page 13: 1 LEMMA 1. dim Rep(¤ ) - dim O = d
Page 17 and 18: PROOF. (2) By inspection the given
Page 19 and 20: §5. Finite representation type In
Page 21 and 22: §6. More homological algebra FROM
Page 23 and 24: LEMMA 3. and - give inverse bijecti
Page 25 and 26: §7. Euclidean case. Preprojectives
Page 27 and 28: LEMMA 4. If ¤ is a positive real r
Page 29 and 30: PROPERTIES. Let X be regular simple
Page 31 and 32: DEFINITION. Given a -orbit of regul
Page 33 and 34: LEMMA 2. If X is regular, X £ 0 th
Page 35 and 36: REMARKS. ~ (1) For the Kronecker qu
Page 37 and 38: (4) Auslander-Reiten theory for wil

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