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

Let $Lambda$ be the set of partitions of length $geq 0$. 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 $mathbb{A}^d_q(Lambda)$ has some kind of quasi-commutativity, and the associativity comes from combinatorial properties of certain polynomials appeared in the quantized cohomological Hall algebra $mathcal{H}^d_q$ of the $d$-loop quiver. It turns out that $mathbb{A}^d_q(Lambda)$ is isomorphic to $mathcal{H}^d_q$, thus can be viewed as a combinatorial realization for $mathcal{H}^d_q$.