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10) is indeed isomorphic to sl3 (C). 3 Intermezzo: quantum groups yielding a six-dimensional complex vector space V . 5. Recall that sln (C) is the Lie algebra of complex n × n-matrices of trace zero. Let eij denote the n × n-matrix which has zeroes everywhere except for a 1 in entry ij. The collection {eij | i = j} ∪ {di := eii − ei+1,i+1 | i = 1, . . , n − 1} is a natural basis of sln (C), and we infer that dimC (sln (C)) = n2 − 1. Note that [eij , ekl ] = δjk eil − δli ekj . Let us now show how to define a linear map ψ : gA −→ sl3 (C) on the generators G in such a way as to make it a Lie algebra homomorphism.

3 Nilpotency & Gabriel’s Theorem In this section, we determine the correct abelian category of quiver representations to associate a Hall algebra to. Furthermore, we consider two classification results of indecomposables in this category: Gabriel’s theorem classifying those of quivers of finite representation type, and Kac’s theorem classifying those of any quiver. 16. In order to associate a Hall algebra to an abelian category, we need the category to be finitary. The category of finite-dimensional representations of a quiver satisfies this requirement, as follows from the previous corollary.

In particular, Rep k (Q0 ) k[X]-Mod . There is an obvious simple representation of Q0 , namely S = (k, 0). More generally however, for any λ ∈ k the representation Sλ = (k, mλ ) where mλ (x) = λx is a simple one. Thus, there are at least as much simple representations as there are elements of k. Let M = (V, f ), N = (W, g) be two representations of Q0 . Choose bases for both V and W , and write f and g in these basis. We denote the corresponding matrices by F and G respectively. Then M = (k m , F ) and N = (k n , G) are isomorphic if and only if n = m and F and G are conjugate matrices.

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An introduction to Hall algebras by Sjoerd Beentjes

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