By Sjoerd Beentjes
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Additional info for An introduction to Hall algebras
Let us now examine under what conditions this map is a (co)algebra morphism. 2, let [M ]kR¯ , [N ]kS¯ be basis elements of HAe . 14) determines the product structure on the extended Hall algebra. Hence G∗e [M ]kR¯ · [N ]kS¯ = (N, R)Am G∗ [M ] · [N ] kG (R)+G (S) ! = (N, R)Am G∗ [M ] · G∗ [N ] kG (R) · kG (S) , where we have used in the second equality that G∗ is a morphism of algebras, whereas G∗e [M ]kR¯ ) · G∗e [N ]kS¯ ) = G∗ [M ]kG (R) · G∗ [N ]kG (S) = (G (N ), G (R))Bm G∗ [M ] · G∗ [N ] kG (R) · kG (S) .
Denote the associated matrix by A = (aij )i,j∈Q0 . It is a symmetric matrix with integer coefficients satisfying aii = 2, aij 0 if i = j, aij = aji . Such a matrix is called a symmetric generalised Cartan matrix. Note that if Q is a connected quiver, this matrix cannot be decomposed into block-diagonal parts (it is irreducible). 5 Note that up to isomorphism, any non-zero scalar will do, since all such representations are isomorphic. 1]. This procedure uses the concept of a realisation of such a matrix, which we describe now.
The matrix of the symmetrised additive Euler form of Q is positive definite; 3. , the associated Kac-Moody algebra g is a simple Lie algebra of type An (n 1), Dm (m 4), or El (l = 6, 7, 8). Moreover, the dimension vector dim establishes a bijection between the set of indecomposable objects of K(Rep k (Q)) and the set ∆+ of positive roots of g. The original proof can be found in . Somewhat later, Bernstein, Gelfand, and Ponomarev proved this result in  by the use of reflection functors. These are a categorification of the reflections in the root system of a simple Lie algebra that generate its Weyl group.