Cohomological Hall algebras and 2 Calabi-Yau categories

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dc.contributor.author Ren, Jie
dc.date.accessioned 2017-08-14T16:54:08Z
dc.date.available 2017-08-14T16:54:08Z
dc.date.issued 2017-08-01 en_US
dc.identifier.uri http://hdl.handle.net/2097/36260
dc.description.abstract The motivic Donaldson-Thomas theory of 2-dimensional Calabi-Yau categories can be induced from the theory of 3-dimensional Calabi-Yau categories via dimensional reduction. The cohomological Hall algebra is one approach to the motivic Donaldson-Thomas invariants. Given an arbitrary quiver one can construct a double quiver, which induces the preprojective algebra. This corresponds to a 2-dimensional Calabi-Yau category. One can further construct a triple quiver with potential, which gives rise to a 3-dimensional Calabi-Yau category. The critical cohomological Hall algebra (critical COHA for short) is defined for a quiver with potential. Via the dimensional reduction we obtain the cohomological Hall algebra (COHA for short) of the preprojective algebra. We prove that a subalgebra of this COHA consists of a semicanonical basis, thus is related to the generalized quantum groups. Another approach is motivic Hall algebra, from which an integration map to the quantum torus is constructed. Furthermore, a conjecture concerning some invariants of 2-dimensional Calabi-Yau categories is made. We investigate the correspondence between the A∞-equivalent classes of ind-constructible 2-dimensional Calabi-Yau categories with a collection of generators and a certain type of quivers. This implies that such an ind-constructible category can be canonically reconstructed from its full subcategory consisting of the collection of generators. en_US
dc.description.sponsorship National Science Fundation en_US
dc.language.iso en_US en_US
dc.publisher Kansas State University en
dc.subject Cohomological Hall algebra en_US
dc.subject 2-dimensional Calabi-Yau category en_US
dc.subject Quiver en_US
dc.subject Semicanonical basis en_US
dc.subject Donaldson-Thomas series en_US
dc.title Cohomological Hall algebras and 2 Calabi-Yau categories en_US
dc.type Dissertation en_US
dc.description.degree Doctor of Philosophy en_US
dc.description.level Doctoral en_US
dc.description.department Department of Mathematics en_US
dc.description.advisor Yan S. Soibelman en_US
dc.date.published 2017 en_US
dc.date.graduationmonth August en_US


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