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By Sjoerd Beentjes

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11. Calculating extensions The fact that there is such a clear geometric description of k Q’s algebraic structure allows for the construction of both projective and injective resolutions of length one of its finite-dimensional modules. We will use these to calculate extensions of representations. To see how this works, let us place path algebras in a somewhat wider context. 12. Let Λ0 be a semi-simple ring, and let Λ1 be a finite length Λ0 -bimodule. The associated tensor ring T (Λ0 , Λ1 ) is the N-graded Λ0 -module Λ := Λr where Λr = Λ1 ⊗Λ0 .

5. Let Q be a quiver, and let M = (Vi , fα ) be a representation of Q. A collection {Wi }i∈Q0 consisting of k-linear subspaces Wi Vi is called a subrepresentation of M if fα (Ws(α) ) ⊆ Wt(α) for all α ∈ Q1 . We call a non-zero representation M simple or irreducible if its only subrepresentations are the trivial representation 0 and M itself. Lastly, such a representation M is called indecomposable if is not isomorphic to a direct sum of two of its non-trivial subrepresentation. 6. Examples Let us reconsider the above four quivers, and see how their representation theory looks like.

Let Q be any quiver, let k be a finite field with ν = |k|, and let A denote the finitary hereditary abelian category of finite-dimensional nilpotent representation of Q over k. 1. Let M be an object of finite length of some abelian category C . Recall that a composition series of M is a filtration 0 = M0 M1 ... Mn−1 Mn = M such that succesive quotients Mi /Mi−1 are simple objects of C . 19 the set S of simple representations of Q is parametrised by its finite set of vertices: S = {S(i) | i ∈ Q0 }.

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An introduction to Hall algebras by Sjoerd Beentjes

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