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Lusztig §4 - Quivers  -  4.1 For a Dynkin quiver Ω, denote by MΩ the (abelian) category of modules (representations) over a fixed field F -  4.2 Any simple module is isomorphic to some \( e_i ≔ V_i = F, V_j = 0 \) -  4.3 Full subcategories \( M_i^+Ω \) and \( M_i^-Ω \) characterized by Hom -  Reflection functors defined when i is a sink (source), so that \( s_iΩ \) has i as a source (sink) -  Equivalence of categories \( Φ_i^+|:M_i^+Ω → M_i^-s_iΩ \) -  4.4 Canonical exact sequence for any V ∈ MΩ -  4.5 Canonical exact sequence splits (always) iff M ≅ M' ⋆ M'' -  In particular, \( V ≅ V(i) ⋆ Φ_i^-Φ_i^+(V) \) -  Corollary: \( Φ_i^-Φ_i^+(V) ≅ V \) iff \( V ∈ M_i^+Ω \) -  The relationship ⋆ is preserved under the equivalence of categories \( M_i^+Ω ≃ M_i^-s_iΩ -  4.6 Assume i a sink of Ω, V ∈ MΩ indecomposable but not isomorphic to \( e_i \) -  Then \( V ∈ M_i^+Ω \) https://workflowy.com/#/005996c4a75c "https://workflowy.com/#/005996c4a75c"  -  {indecomposables of \( M_i^+Ω \) other than \( e_i \)} ↔ { indecomposables of \( M_i^-s_iΩ \) other than \( e_i \)} -  4.7-4.11 skip because it's mostly about representations -  Prop 4.12 For a fixed orientation Ω of an ADE Dynkin quiver, TFAE: Needed "Needed  only to prove 4.13 https://workflowy.com/#/26775c0447fc https://workflowy.com/#/26775c0447fc"  -  This proof is fucking bizarre -  First he shows (b) ⇒ (a) and (b) ⇒ (c). Then he shows that (a) & (c) ⇒ (b). We still need (a) ⇒ (b) and (c) ⇒ (b), which he does by first assuming that all of (a), (b), (c) hold for a larger graph which has "our" rank n graph as a subgraph. Since (a-large) ⇒ (a-small) and (c-large) ⇒ (c-small), so (a-large) & (c-large) ⇒ (a-small) & (c-small) ⇒ (b-small) ⇒ (a-small), (c-small). -  Since we can choose the embedding (small) ⊆ (large) such that w₀(large) is central in W(large), that is, xw₀ = w₀x for all x ∈ W(large) -  That is, we reduce the general case to the case where w₀ is central in W (for the subgraph) -  Now show that if w₀ is central in W, then (b) holds, and hence (a) and (c) hold as well -  The proof of this uses the Coxeter element, screw it -  The point seems to be that it is possible to unwind the sequence of \( s_{i_j} \) applied to the roots because each \( s_{i_j} \) is self-inverse -  In particular, the order of the roots is cyclic -  There exists some reduced expression of w₀ adapted to Ω -  (2) Gabriel's Theorem + existence of a total order on R⁺ such that \( \textrm{Hom}(e_{α^k},e_{α^{k'}}) = 0 \) whenever \( α^{k'} < α^{k} \) -  (2) ⇒ (1) -  If \( α^1<…<α^ℓ \) is such an order, then \( α^1 = α_{i_1} \) for some \( i_1 \) which is a sink of Ω -  \( e_{α^1} \) is simple, hence is isomorphic to some \( e_{α_i} \) -  \( \textrm{Hom}(e_{α^k},e_{α_{i_1}}) = 0 \) for all \( k ≠ i_1 \) implies \( i_1 \) is a sink -  Argument involving the Coxeter element -  Choose the embedding of the smaller Dynkin diagram into the larger Dynkin diagram to be such that w₀ is central (in which Weyl group -  OK, this means that conjugation by w₀ is the identity -  But we know that (at least in type A) conjugation by w₀ is never central (except for A₁), so WTF? -  OK, anyway, define a Coxeter element \( c = s_{i_1}s_{i_2}⋯s_{i_n} ∈ W \) where \( i_j ← i_{j'} \) implies j < j' -  Then the order of c is even -   -  4.13 There is a 1-1 correspondence {reduced expressions for w₀ adapted to Ω} ↔ {total orders on R⁺ such that \( \textrm{Hom}(e_{α^k},e_{α^{k'}}) = 0 \) whenever \( α^{k'} < α^{k} \)} -  4.14(a) If i is a Ω-adapted w₀-redex, then \( (i_2,…,i_ℓ, j) \) where \( s_j ≔ w_0 s_{i_1} w_0 \) is a \( s_{i_1}Ω \)-adapted w₀-redex -  4.14(b) If i ← j in Ω, then \( α_i < α_j \) -