By Sjoerd Beentjes
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Additional info for An introduction to Hall algebras
Given a representation M = (Vi , fα ) of Q, the dimension vector of M is the element dim(M ) := (dim Vi )i∈Q0 of ZQ0 . Note that if 0 → M → M → M → 0 is a short exact sequence of such representations, we have dim(M ) = dim(M )+dim(M ). 1) dim : K (rep k (Q)) → ZQ0 on the Grothendieck group of rep k (Q). We will later identify ZQ0 with the root lattice of the Kac-Moody algebra associated to the undirected graph Q underlying Q. Before giving some examples, let us first introduce two important representation-theoretic definitions.
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 . 14) determines the product structure on the extended Hall algebra. Hence G∗e [M ]kR¯ · [N ]kS¯ = (N, R)Am G∗ [M ] · [N ] kG (R)+G (S) ! = (N, R)Am G∗ [M ] · G∗ [N ] kG (R) · kG (S) , where we have used in the second equality that G∗ is a morphism of algebras, whereas G∗e [M ]kR¯ ) · G∗e [N ]kS¯ ) = G∗ [M ]kG (R) · G∗ [N ]kG (S) = (G (N ), G (R))Bm G∗ [M ] · G∗ [N ] kG (R) · kG (S) .
Remark. Two such realisations (h1 , Π1 , Π∨ 1 ), (h2 , Π2 , Π2 ) are called isomorphic if there exists ∨ ∗ a complex vector space isomorphism φ : h1 → h2 such that φ(Π∨ 1 ) = Π2 and φ (Π2 ) = Π1 . By [33, Prop. 1], any two realisations of A are isomorphic. 3. Example Take as quiver Q = • → • = A2 , of which the undirected graph is the Dynkin diagram of the simple Lie algebra sl3 (C). Its associated Cartan matrix is 2 −1 . 10) Note that rk(A) = 2 and s = 2, so we are looking for a complex vector space h of dimension dimC (h) = 2.