By Sjoerd Beentjes
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Additional resources for An introduction to Hall algebras
Note that addition is formal, not induced by direct summation in A. In order to have G∗ be a morphism of algebras or coalgebras, something more is required. Let [M ], [N ] ∈ Iso(A), then we see that G∗ ([M ] · [N ]) = M, N A m R FM,N [G (R)], [R]∈Iso(A) whilst G∗ ([M ]) · G∗ ([N ]) = G (M ), G (N ) B m FGS(M ),G (N ) [S]. [S]∈Iso(B) Therefore these two expressions are equal if G preserves the Euler-form and if it preserves the R sets FM,N for all objects M, N, R of A. Equivalently, this is so if G is an exact functor and ∼ if it is extremely faithful: it induces isomorphisms ExtiA (M, N ) −→ ExtiB (G (M ), G (N )) for all M, N ∈ Ob(A), for all i 0.
Of tensors under the identification kQr = 0 0 The above projective resolution translates into the following somewhat more explicit statement, originally due to Ringel. 13 (Ringel). Let M be a left k Q-module. Then the following sequence v u P (i) ⊗k ei M −→ M −→ 0 P (t(α)) ⊗k es(α) M −→ 0 −→ i∈Q0 α∈Q1 is a projective resolution of k Q-modules. The maps u = α∈Q1 uα and v = i∈Q0 vi are defined as uα (p ⊗k m) = pα ⊗k m − p ⊗k α · m, and vα (q ⊗k n) = q · n, where p ∈ P (t(α)), m ∈ et(α) M , and q ∈ P (i), m ∈ ei M .
Vn−1 −→ Vn . To each vertex i ∈ Q0 corresponds a simple representation Si of An with zero maps and vector spaces Vj = kδij ; these will turn out to be all simple representations of An . 2] that classifies all indecomposable representations of An . This is roughly done by considering kernels and images of the linear maps of the representation, and showing every non-zero map αi must be the identity map ✶ : Vi → Vi+1 . Once more by indecomposablility, every non-zero vector space is found to be k.