I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-symmetrization leads to the concept of cohomological Hall algebra. My question is about the "conventional" as he calls it symmetrization. In specific I would like to understand the following paragraph:

Apparently by symmetrizing the Euler-Ringel form $\chi_Q$ we can associate to the quiver a Kac-Moody algebra $y_Q$ and make a triangular decomposition to positive, Cartan and negative parts $$ y_Q = y_Q^{+} \oplus \mathfrak{h}_Q \oplus y_Q^-. $$ Then by considering quivers over a finite field $\mathbb{F}_q$ we can define the Hall algebra $H_Q$ which is the algebra of function with finite support on the (moduli) stack of the abelian category of representations of $Q$. For two functions (they have compact support so maybe they are Dirac-delta type functions) $f_1, f_2$ we have $$ (f_1 \cdot f_2)(E) = \# \sum_{0\to E_1 \to E \to E_2 \to 0}f_1(E_1) \cdot f_2(E_2) $$ i.e., we sum over all possible extensions of the rep. $E$. Then, this Hall algebra $H_Q$ is somehow related with the quantum group $\mathcal{U}_q(y_Q^+)$.

Furthermore, in physics this constructible Hall algebra appears in what Soibelman calls "instanton side" of geometric engineering (I guess he means gauge theoretic side). By considering rank $r < \infty$ torsion free sheaves on $\mathbb{CP}^2$ (with some framing). These sheaves have a moduli space $\mathcal{M}_n$ with $n=ch_2(E)$ for sheaf $E$. Apparently there is some Hecke operators entering here (how?).

Soibelman claims that these are standard conventional stuff (known before the notion of cohomological Hall algebras etc). He mentions Nakajima, Schiffmann, Okounkov, Kapranov, ...

I want to have a precise understanding of these "conventional" constructible Hall algebra stuff. Could you explain in more precise terms these previous two paragraphs or could you provide me with a reference(s) that explain these constructions and relations?



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