By Sjoerd Beentjes
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Extra info for An introduction to Hall algebras
I∈Q0 One verifies that the association M → F (M ) defines a covariant functor Rep k (Q) → Mod k Q. Conversely, given a left k Q-module M we define (GM )i := ei · M , and (GM )α : (GM )s(α) → (GM )t(α) , m → α · m. Let φ : M → N be a morphism, then Gφ : GM → GN is defined by restriction in the following sense. For m ∈ (GM )i = ei · M we have φ(m) = φ(ei · m) = ei · φ(m) ∈ ei · N = (GN )i . One now verifies easily that GF = ✶ on Rep k (Q) and GF ✶ on Mod k Q. 9. Let M , N , E be representations of Q.
Let φ : M → N be a morphism, then Gφ : GM → GN is defined by restriction in the following sense. For m ∈ (GM )i = ei · M we have φ(m) = φ(ei · m) = ei · φ(m) ∈ ei · N = (GN )i . One now verifies easily that GF = ✶ on Rep k (Q) and GF ✶ on Mod k Q. 9. Let M , N , E be representations of Q. We call E an extension of M by N if there exists a short exact sequence 0 → N → E → M → 0. 5 applies. Thus we can alternatively think of extensions as elements of Ext1kQ (M, N ) and calculate them using techniques of homological algebra.
1 |x); note that it runs from right to left. Each vertex x ∈ Q0 defines a path (x|x) of length zero, each arrow α ∈ Q1 defines a path (t(α)|α|s(α)) of length one. 7. , s(p) = t(q), and zero otherwise. Note that 1 = i ei is the unit of this associative k-algebra since the set of vertices Q0 is assumed to be finite. Furthermore, let Qr denote the set of paths of length r ∈ N0 , which extends the notation for vertices Q0 and arrows Q1 , and let kQn be the k-vector space with basis Qn . There is a decomposition of the path algebra as kQ = kQn where kQr · kQs = kQr+s n 0 by construction, so k Q is a N0 -graded algebra.