Given a quiver Q with/without potential, one can construct an algebra structure on the

cohomology of the moduli stacks of representations of Q. The algebra is called Cohomological Hall algebra (COHA for short). One can also add a framed structure to quiver Q, and

discuss the moduli space of the stable framed representations of Q. Through these geometric constructions, one can construct two representations of Cohomological Hall algebra of Q

over the cohomology of moduli spaces of stable framed representations. One would get the

double of the representations of Cohomological Hall algebras by putting these two representations together. This double construction implies that there are some relations between

Cohomological Hall algebras and some other algebras.

In this dissertation, we focus on the quiver without potential case. We first define Cohomological Hall algebras, and then the above construction is stated under some assumptions.

We computed two examples in detail: A₁-quiver and Jordan quiver. It turns out that A₁-

COHA and its double representations are related to the half infinite Clifford algebra, and

Jordan-COHA and its double representations are related to the infinite Heisenberg algebra.

Then by the fact that the underlying vector spaces of these two COHAs are isomorphic to

each other, we get a COHA version of Boson-Fermion correspondence.