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Cluster structures for 2-Calabi-Yau categories and unipotent groups
We investigate cluster tilting objects (and subcategories) in triangulated
2-Calabi-Yau categories and related categories. In particular we construct a
new class of such categories related to preprojective algebras of non Dynkin
quivers associated with elements in the Coxeter group. This class of
2-Calabi-Yau categories contains the cluster categories and the stable
categories of preprojective algebras of Dynkin graphs as special cases. For
these 2-Calabi-Yau categories we construct cluster tilting objects associated
with each reduced expression. The associated quiver is described in terms of
the reduced expression. Motivated by the theory of cluster algebras, we
formulate the notions of (weak) cluster structure and substructure, and give
several illustrations of these concepts. We give applications to cluster
algebras and subcluster algebras related to unipotent groups, both in the
Dynkin and non Dynkin case.Comment: 49 pages. For the third version the presentation is revised,
especially Chapter III replaces the old Chapter III and I
Differential-algebraic jet spaces preserve internality to the constants
This paper concerns the model theory of jet spaces (i.e., higher-order
tangent spaces) in differentially closed fields. Suppose p is the generic type
of the jet space to a finite dimensional differential-algebraic variety at a
generic point. It is shown that p satisfies a certain strengthening of almost
internality to the constant field called "preserving internality to the
constants". This strengthening is a model-theoretic abstraction of the generic
behaviour of jet spaces in complex-analytic geometry. A counterexample is
constructed showing that only this generic analogue holds in
differential-algebraic geometry.Comment: 13 page
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