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General representations of quivers. (English) Zbl 0795.16008
General representations of a quiver $$Q$$ are investigated in the paper. Recall that the representations of a dimension vector $$\alpha$$ of the quiver $$Q$$ are parametrised by a vector space $$R(Q,\alpha)$$ on which an algebraic group $$\text{Gl}(Q,\alpha)$$ acts such that the orbits of the group on the space are in 1-1 correspondence with the isoclasses of representations. If $$R_ p$$ is the representation associated to the point $$p$$ and $$R_ p= \oplus S_{i,p}$$ with $$S_{i,p}$$ indecomposable, V. Kac proved that there is an open subset $$U$$ of $$R(Q,\alpha)$$ such that for all $$p\in U$$, the set of dimension vectors $$\{\underline{\dim} S_{i,p}\}$$ is independent of $$p$$; he calls the sum $$\alpha= \sum \underline{\dim} S_{i,p}$$ the canonical decomposition of $$\alpha$$. In the reviewed paper an algorithm is provided to compute the canonical decomposition of an arbitrary dimension vector in terms of the Euler form of the quiver and to compute the dimension vectors of subrepresentations of a general representation and the minimal dimension of $$\text{Ext}(R,S)$$ as $$R$$ and $$S$$ run through representations of dimension vectors $$\alpha$$ and $$\beta$$ respectively.

##### MSC:
 16G20 Representations of quivers and partially ordered sets
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