Cohomological Hall algebras and 2 Calabi-Yau categories
Doctor of Philosophy === Department of Mathematics === Yan S. Soibelman === 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 th...
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ndltd-KSU-oai-krex.k-state.edu-2097-362602017-08-20T03:41:02Z Cohomological Hall algebras and 2 Calabi-Yau categories Ren, Jie Cohomological Hall algebra 2-dimensional Calabi-Yau category Quiver Semicanonical basis Donaldson-Thomas series Doctor of Philosophy Department of Mathematics Yan S. Soibelman 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. 2017-08-14T16:54:08Z 2017-08-14T16:54:08Z 2017 August Dissertation http://hdl.handle.net/2097/36260 en_US Kansas State University |
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en_US |
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Cohomological Hall algebra 2-dimensional Calabi-Yau category Quiver Semicanonical basis Donaldson-Thomas series |
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Cohomological Hall algebra 2-dimensional Calabi-Yau category Quiver Semicanonical basis Donaldson-Thomas series Ren, Jie Cohomological Hall algebras and 2 Calabi-Yau categories |
description |
Doctor of Philosophy === Department of Mathematics === Yan S. Soibelman === 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. |
author |
Ren, Jie |
author_facet |
Ren, Jie |
author_sort |
Ren, Jie |
title |
Cohomological Hall algebras and 2 Calabi-Yau categories |
title_short |
Cohomological Hall algebras and 2 Calabi-Yau categories |
title_full |
Cohomological Hall algebras and 2 Calabi-Yau categories |
title_fullStr |
Cohomological Hall algebras and 2 Calabi-Yau categories |
title_full_unstemmed |
Cohomological Hall algebras and 2 Calabi-Yau categories |
title_sort |
cohomological hall algebras and 2 calabi-yau categories |
publisher |
Kansas State University |
publishDate |
2017 |
url |
http://hdl.handle.net/2097/36260 |
work_keys_str_mv |
AT renjie cohomologicalhallalgebrasand2calabiyaucategories |
_version_ |
1718517057195081728 |