论文标题
$ \ imath $量子组的新实现通过$δ$ -HALL代数
New realization of $\imath$quantum groups via $Δ$-Hall algebras
论文作者
论文摘要
对于本质上是一个小的遗传性阿贝尔类别$ \ MATHCAL {a} $，我们定义了一种新型的代数$ \ Mathcal {h}_Δ（\ Mathcal {a}）$，称为$Δ$-Δ$ -HALL代数为$ \ Mathcal {a a} $。 $ \ MATHCAL {H}_δ（\ Mathcal {a}）$的基础是$ \ Mathcal {a} $中的对象类别类别，而$δ$ -HALL数字计算某些在$ \ Mathcal {A} $中的精确序列的三个序列。我们表明，$δ$ -HALL代数$ \ MATHCAL {h}_Δ（\ Mathcal {a}）$与$ \ Mathcal {a a} $的1个周期派生的霍尔代数同构。通过进行合适的扩展和扭曲，我们可以分别获得$ \ imath $ hall代数和与$ \ Mathcal {a} $相关的半衍生霍尔代数。 当应用于nilpotent表示类别时，$ \ MATHCAL {a} = {\ rm rep^{nil}}（\ MathBf {k} q）n intunary Quiver $ q $ n notary Quiver $ q $ n notal loops n nolop loops，the（\ emph {desp。 $ \ imath $量子组与$ q $相关。
For an essentially small hereditary abelian category $\mathcal{A}$, we define a new kind of algebra $\mathcal{H}_Δ(\mathcal{A})$, called the $Δ$-Hall algebra of $\mathcal{A}$. The basis of $\mathcal{H}_Δ(\mathcal{A})$ is the isomorphism classes of objects in $\mathcal{A}$, and the $Δ$-Hall numbers calculate certain three-cycles of exact sequences in $\mathcal{A}$. We show that the $Δ$-Hall algebra $\mathcal{H}_Δ(\mathcal{A})$ is isomorphic to the 1-periodic derived Hall algebra of $\mathcal{A}$. By taking suitable extension and twisting, we can obtain the $\imath$Hall algebra and the semi-derived Hall algebra associated to $\mathcal{A}$ respectively. When applied to the the nilpotent representation category $\mathcal{A}={\rm rep^{nil}}(\mathbf{k} Q)$ for an arbitrary quiver $Q$ without loops, the (\emph{resp.} extended) $Δ$-Hall algebra provides a new realization of the (\emph{resp.} universal) $\imath$quantum group associated to $Q$.
