By Sjoerd Beentjes

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18) between 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.

Fα1 πt(α1 ) (m) and by extending k-linearly. So the action of a path projects down the element m to the vector space at the starting vertex of the path, chases the element through all consecutive arrows of the quiver, then includes the result back in F M . Given a morphism θ : M → N of representations of Q, we define F θ = i∈Q0 θi . This is a k Q-module morphism since for α ∈ Q1 , m ∈ F M we find M F θ(α · m) = ιN i θi fα πi (m) M = ιN i gα θi πi (m) i∈Q0 = α · F θ(m). i∈Q0 One verifies that the association M → F (M ) defines a covariant functor Rep k (Q) → Mod k Q.

DimC (h) = 2s − rk(A); 3. αi (hj ) = aji for all i, j = 1, 2, . . , 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 .

### An introduction to Hall algebras by Sjoerd Beentjes

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