12 research outputs found
From the potential to the first Hochschild cohomology group of a cluster tilted algebra
The objective of this paper is to give a concrete interpretation of the
dimension of the first Hochschild cohomology space of a cyclically oriented or
tame cluster tilted algebra in terms of a numerical invariant arising from the
potential
On the first Hochschild cohomology group of a cluster-tilted algebra
Given a cluster-tilted algebra B, we study its first Hochschild cohomology
group HH^1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra
such that B is the relation extension of C, then we show that if C is
constrained, or else if B is tame, then HH^1(B) is isomorphic, as a k-vector
space, to the direct sum of HH^1(C) with k^{n\_{B,C}}, where n\_{B,C} is an
invariant linking the bound quivers of B and C. In the representation-finite
case, HH^1(B) can be read off simply by looking at the quiver of B.Comment: 30 page
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Representation Theory of Quivers and Finite Dimensional Algebras
Methods and results from the representation theory of quivers and
finite dimensional algebras have led to many interactions with other
areas of mathematics. Such areas include the theory of Lie algebras
and quantum groups, commutative algebra, algebraic geometry and
topology, and in particular the theory of cluster algebras. The aim of
this workshop was to further develop such interactions and to
stimulate progress in the representation theory of algebras
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Developments in the mathematics of the A-model: constructing Calabi-Yau structures and stability conditions on target categories
This dissertation is an exposition of the work conducted by the author in the later years of graduate school, when two main projects were completed. Both projects concern the application of sheaf-theoretic techniques to construct geometric structures on categories appearing in the mathematical description of the A-model, which are of interest to symplectic geometers and mathematicians working in mirror symmetry. This dissertation starts with an introduction to the aspects of the physics of mirror symmetry that will be needed for the exposition of the techniques and results of these two projects. The first project concerns the construction of Calabi-Yau structures on topological Fukaya categories, using the microlocal model of Nadler and others for these categories. The second project introduces and studies a similar local-to-global technique, this time used to construct Bridgeland stability conditions on Fukaya categories of marked surfaces, extending some results of Haiden, Katzarkov and Kontsevich on the relation between stability of Fukaya categories and geometry of holomorphic differentials
International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022
Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022.
Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress.
The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library