Ren, Jie2017-08-142017-08-142017-08-01http://hdl.handle.net/2097/36260The 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© the author. This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).http://rightsstatements.org/vocab/InC/1.0/Cohomological Hall algebra2-dimensional Calabi-Yau categoryQuiverSemicanonical basisDonaldson-Thomas seriesDissertation