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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 \( α_i < α_j \)-