Cohomological Hall algebras and 2 Calabi-Yau categories

Date

2017-08-01

Authors

Journal Title

Journal ISSN

Volume Title

Publisher

Kansas State University

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.

Description

Keywords

Cohomological Hall algebra, 2-dimensional Calabi-Yau category, Quiver, Semicanonical basis, Donaldson-Thomas series

Graduation Month

August

Degree

Doctor of Philosophy

Department

Department of Mathematics

Major Professor

Yan S. Soibelman

Date

Type

Dissertation

Citation