By Sjoerd Beentjes

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**Additional info for An introduction to Hall algebras**

**Sample text**

When G is exact. g. 6]). Consequently, we obtain a morphism of algebras between the associated commutative group algebras, which we shal denote by C[K (G )] : C[K (A)] −→ C[K (B)]. The previous discussions shows that the existence of an exact functor G : A → B between two finitary abelian categories induces a well-defined linear map G∗e := G∗ ⊗ C[K (G )] : HAe −→ HBe of vector spaces. Let us now examine under what conditions this map is a (co)algebra morphism. 2, let [M ]kR¯ , [N ]kS¯ be basis elements of HAe .

The parameter v is a formal variable, which we want to specialize at prime powers q = pn . But the Hopf algebra is defined over the field C(v), so a specialization at a value ν ∈ C∗ need not be well defined. Lusztig presented a solution which we will present shortly. 2. Giving the generators Ei , Fi , v h weights αi , −αi , 0 respectively equips Uv (g) with a grading by the root lattice R = Z · αi of g. Note that the Hall algebra associated to Q is graded by the Grothendieck group of A = Rep k (Q).

We allow loops and multi-edges. 1 Generalities on quivers and s, t : Q1 → Q0 are functions assigning to an edge its source and tail vertex respectively. The undirected graph underlying a quiver Q will be denoted by Q = (Q0 , Q1 ). So if α : i → j is an edge from vertex i to vertex j in some quiver Q, then s(α) = i and t(α) = j. We will denote by Qij = {α ∈ Q1 | s(α) = i, t(α) = j} the set of edges going from i to j in Q. 2. Examples Let us consider four examples of quivers, that will later either illustrate the different possible complexities of their representation theory or appear in an application.