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Strong exceptional sequences provided by quivers. (English) Zbl 0951.16006
Let \(Q\) be a connected quiver without oriented cycles. It consists of a set \(Q_0\) of vertices, a set \(Q_1\) of arrows, and two functions \(s,t\colon Q_1\to Q_0\) assigning to each arrow \(\alpha\in Q_1\) its source \(s(\alpha)\) and target \(t(\alpha)\). A representation of \(Q\) is a collection of finite-dimensional \(k\)-vector spaces \(x(q)\) for each vertex \(q\) together with a collection of linear maps \(x(\alpha)\colon x(s(\alpha))\to x(t(\alpha))\), \(\alpha\in Q_1\). A representation is called thin sincere if \(\dim_kx(q)=1\) for all \(q\in Q_0\). Let \({\mathcal U}\to{\mathcal M}(Q)\) be the fine moduli space of stable thin sincere representations of \(Q\) with respect to the canonical stability notion [A. King, Q. J. Math., Oxf. II. Ser. 45, No. 180, 515-530 (1994; Zbl 0837.16005)]. The authors prove that the universal bundle \(\mathcal U\) has no self-extensions, that is, \(\text{Ext}^\ell_{{\mathcal M}(Q)}({\mathcal U},{\mathcal U})=0\) for all \(\ell>0\), and compute the endomorphism algebra of the universal bundle \(\mathcal U\). They then obtain a necessary and sufficient condition under which this algebra is isomorphic to the path algebra of the quiver \(Q\).

16G20 Representations of quivers and partially ordered sets
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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