Caldero-Chapoton algebras

Motivated by the representation theory of quivers with potentials introduced by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave explicit formulae for the cluster variables of Dynkin quivers, we associate a Caldero-Chapoton algebra to any (possibly infinite dimensional) basic algebra. By definition, the Caldero-Chapoton algebra is (as a vector space) generated by the Caldero-Chapoton functions of the decorated representations of the basic algebra. The Caldero-Chapoton algebra associated to the Jacobian algebra of a quiver with potential is closely related to the cluster algebra and the upper cluster algebra of the quiver. The set of generic Caldero-Chapoton functions, which conjecturally forms a basis of the Caldero-Chapoton algebra) is parametrized by the strongly reduced components of the varieties of representations of the Jacobian algebra and was introduced by Geiss, Leclerc and Schr\"oer. Plamondon parametrized the strongly reduced components for finite-dimensional basic algebras. We generalize this to arbitrary basic algebras. Furthermore, we prove a decomposition theorem for strongly reduced components. Thanks to the decomposition theorem, all generic Caldero-Chapoton functions can be seen as generalized cluster monomials. As another application, we obtain a new proof for the sign-coherence of g-vectors. Caldero-Chapoton algebras lead to several general conjectures on cluster algebras.Comment: 35 pages. v2: Corrected the definition of a generic E-invariant and also a short list of minor inaccuracies and typos. Final version to appear in Transactions AM

Über die Köpfe hinweg? Überlegungen, Einwände und Thesen aus der Perspektive Studierender

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Religiöses Bekenntnis und politisches Interesse : Bamberger Hegelwoche 2002

Religiöses Bekenntnis und politisches Interesse : Bamberger Hegelwoche 200

Compact Kaehler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kaehler quotient. Additionally, as a byproduct of our discussion we give an example of a complete Kaehlerian non-projective algebraic surface, which may be of independent interest.Comment: 33 pages, 1 figure; improved exposition, many of the results are now proven for complete and not only for projective quotients, examples showing the necessity of the assumptions made in the main results added; to appear in Advances in Mathematic