An introduction to Hall algebras by Sjoerd Beentjes

By Sjoerd Beentjes

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A morphism u : (V, f ) → (W, g) of representations is a linear map such that gu = uf : it intertwines f and g. Clearly, giving such a pair (V, f ) is equivalent to equipping V with a k[X]-module structure where the action of X on an element v ∈ V is given by X · v = f (v). 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.

In total, we find ν m1 n1 +m2 n2 −m1 n2 {[I]k(m1 ,m2 ) , [J]k(n1 ,n2 ) }e = δIJ ν2 − 1 where I, J are indecomposable representations of A2 , so S1 , S2 or I12 , and mi , ni ∈ Z. 3 Intermezzo: quantum groups In this short intermezzo we describe how to obtain the Kac-Moody algebra associated to a quiver without loops. Furthermore, we give a presentation of the quantised universal enveloping algebra of a simply-laced Kac-Moody Lie algebra in terms of generators and relations; this is a quantum group.

Convention. From now on, we will only consider finite-dimensional nilpotent representations of quivers without loops. Instead of sticking to the more correct rep nil k (Q) for this category, we will slightly abuse notation and simply write A = Rep k (Q). In summary, this is a k-linear finitary hereditary abelian Krull-Schmidt category. In particular, the dimension vector dim : K (Rep k Q) −→ R, M → dim(M ) = dim(Mi )αi i∈Q0 establishes an isomorphism of Z-modules mapping the symmetrised Euler form (−, −)a onto the Cartan-Killing form of the root lattice R of the Kac-Moody algebra associated to Q.

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