重新讨论$d$ -圈的量子化上同调Hall代数
Quantized cohomological Hall algebra of the $d$-loop quiver revisited
摘要：
令$Λ$为长度为$\geq 0$的分区的集合。我们引入了一个与$Λ$相关的$\mathbb{N}$分次代数$\mathbb{A}_q^d(Λ)$，它可以看作是Reineke定义的分块代数的一个量子化. $\mathbb{A}^d_q(Λ)$的乘法运算具有某种拟交换性，而这种结合性则来源于$d$ -环图的量子化上同调Hall代数$\mathcal{H}^d_q$中某些多项式的组合性质.结果表明，$\mathbb{A}^d_q(Λ)$与$\mathcal{H}^d_q$同构，可以看作是$\mathcal{H}^d_q$的组合实现.
英文摘要：
Let $Λ$ be the set of partitions of length $\geq 0$. We introduce an $\mathbb{N}$-graded algebra $\mathbb{A}_q^d(Λ)$ associated to $Λ$, which can be viewed as a quantization of the algebra of partitions defined by Reineke. The multiplication of $\mathbb{A}^d_q(Λ)$ 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 $\mathbb{A}^d_q(Λ)$ is isomorphic to $\mathcal{H}^d_q$, thus can be viewed as a combinatorial realization for $\mathcal{H}^d_q$.
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