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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 \(Q\).

MSC:
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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