By Sjoerd Beentjes
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Extra resources for An introduction to Hall algebras
Thus the representation theory of Z4 encodes the four subspaces problem in Linear Algebra, alluded to in the introduction. This is a wild problem, which can be seen as follows. Let us restrict our attention to indecomposable representations M of dimension vector dim(M ) = (1, 1, 1, 1, 2). 1 Generalities on quivers equivalences. These equivalences are encoded in an action of the projective linear group PGL(2, k), so the isomorphism classes of indecomposable representations of Z4 correpond bijectively to the orbits of PGL(2, k) on P1 (k) × P1 (k) × P1 (k) × P1 (k); there are infinitely many orbits, provided the ground field k is infinite.
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.
Just like in the finite type case, one can construct a generic Hall algebra in the tame case. This has recently been done by Hubery . 1 Generalities on quivers elements of the ground field. For this, it is important that the classification of indecomposable modules does not depend on the ground field. The nilpotent category of both the Jordan quiver and the cyclic quiver are independent of the chosen ground field k = Fq . This is not true however in the non-nilpotent case, which can be seen as follows.
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