Quantized cohomological Hall algebra of the $d$-loop quiver revisited

Abstract

Let $\Lambda$ be the set of partitions of length $\ge 0$. We introduce an $\mathbb{N}$-graded algebra ${\mathbb{A}}_q^d(\Lambda )$ associated to $\Lambda$, which can be viewed as a quantization of the algebra of partitions defined by Reineke. The multiplication of ${\mathbb{A}}_q^d(\Lambda )$ has some kind of quasi-commutativity, and the associativity comes from combinatorial properties of certain polynomials appeared in the quantized cohomological Hall algebra ${\mathcal{H}}_q^d$ of the $d$-loop quiver. It turns out that ${\mathbb{A}}_q^d(\Lambda )$ is isomorphic to ${\mathcal{H}}_q^d$, thus can be viewed as a combinatorial realization for ${\mathcal{H}}_q^d$.