By Sjoerd Beentjes

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4. Let A, B be two finitary abelian categories, and let G : A −→ B be an exact, extremely faithful functor. Then 1. G∗e : HAe −→ HBe is an embedding of algebras; 2. if in addition A, B are hereditary and G (A) is essentially stable under taking subobjects in B, then G∗e is a morphism of coalgebras too, whence a bialgebra morphism. 5. Exact subcategories Since this will be our main usage of the theory of this section, we specialize the previous results to the case of the inclusion of an exact subcategory.

4]. This means that S(xy) = S(y)S(x) for any two elements x, y of the Hopf algebra. In the case at hand, S([Si ]) = −ki−1 [Si ] since there is only the trivial strict filtration 0 Si for i = 1, 2. Recall that ki−1 ≡ kS−1 = k−Si in C[K (A)]. We find for example i S([S1 ⊕ S2 ]) = S([S1 ])S([S2 ]) = ν −2 kS−1⊕S [S1 ⊕ S2 ] + [I12 ] 1 2 but on the other hand also S([I12 ]) = νS([S2 ])S([S1 ]) − S([S1 ])S([S2 ]) = −kI−1 [I12 ] + (1 − ν −2 )kS−1⊕S [S1 ⊕ S2 ]. 3 Intermezzo: quantum groups Note that the cardinality of the ground field again acts as deformation parameter.

3) where M (λ ⊗ m) = λ · m, and δM ((λ ⊗ µ) ⊗ m) = (λ ⊗ µ) ⊗ m − λ ⊗ (µ · m). Note that we interpret Λ+ as Λ ⊗Λ0 Λ1 here. 1 Generalities on quivers How does this apply to path algebras? Given a quiver Q, we find by a straightforward argument that the vertices ei form a complete set of primitive3 orthogonal idempotents, which implies that kQ0 ∼ = i kei as k-algebras. So kQ0 is semi-simple and we may set Λ0 = kQ0 . By restricting multiplication in the path algebra, kQ1 obtains a kQ0 -bimodule structure.

### An introduction to Hall algebras by Sjoerd Beentjes

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