We prove that a subalgebra of this COHA consists of a semicanonical basis, thus is related to the generalized quantum groups. Via the dimensional reduction we obtain the cohomological Hall algebra (COHA for short) of the preprojective algebra. The critical cohomological Hall algebra (critical COHA for short) is defined for a quiver with potential. One can further construct a triple quiver with potential, which gives rise to a 3-dimensional Calabi-Yau category. This corresponds to a 2-dimensional Calabi-Yau category. Given an arbitrary quiver one can construct a double quiver, which induces the preprojective algebra. The cohomological Hall algebra is one approach to the motivic Donaldson-Thomas invariants. 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.
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