%0 Generic %1 chen2022realization %A Chen, Jiayi %A Lin, Yanan %A Ruan, Shiquan %D 2022 %K Hall algebras %T New realization of $ımath$quantum groups via $\Delta$-Hall algebras %U http://arxiv.org/abs/2209.00205 %X For an essentially small hereditary abelian category $A$, we define a new kind of algebra $H_\Delta(A)$, called the $\Delta$-Hall algebra of $A$. The basis of $H_\Delta(A)$ is the isomorphism classes of objects in $A$, and the $\Delta$-Hall numbers calculate certain three-cycles of exact sequences in $A$. We show that the $\Delta$-Hall algebra $H_\Delta(A)$ is isomorphic to the 1-periodic derived Hall algebra of $A$. By taking suitable extension and twisting, we can obtain the $ımath$Hall algebra and the semi-derived Hall algebra associated to $A$ respectively. When applied to the the nilpotent representation category $A=\rm rep^nil(k Q)$ for an arbitrary quiver $Q$ without loops, the (resp. extended) $\Delta$-Hall algebra provides a new realization of the (resp. universal) $ımath$quantum group associated to $Q$.

@misc{chen2022realization, abstract = {For an essentially small hereditary abelian category $\mathcal{A}$, we define a new kind of algebra $\mathcal{H}_{\Delta}(\mathcal{A})$, called the $\Delta$-Hall algebra of $\mathcal{A}$. The basis of $\mathcal{H}_{\Delta}(\mathcal{A})$ is the isomorphism classes of objects in $\mathcal{A}$, and the $\Delta$-Hall numbers calculate certain three-cycles of exact sequences in $\mathcal{A}$. We show that the $\Delta$-Hall algebra $\mathcal{H}_{\Delta}(\mathcal{A})$ 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) $\Delta$-Hall algebra provides a new realization of the (\emph{resp.} universal) $\imath$quantum group associated to $Q$.}, added-at = {2022-09-02T16:05:25.000+0200}, author = {Chen, Jiayi and Lin, Yanan and Ruan, Shiquan}, biburl = {https://www.bibsonomy.org/bibtex/243f98889e33f56a1e7a66ce0d06a5d4f/dragosf}, description = {New realization of $\imath$quantum groups via $\Delta$-Hall algebras}, interhash = {b444b78f76c37b984526cd91ab39691b}, intrahash = {43f98889e33f56a1e7a66ce0d06a5d4f}, keywords = {Hall algebras}, note = {cite arxiv:2209.00205Comment: 19 pages}, timestamp = {2022-09-02T16:05:25.000+0200}, title = {New realization of $\imath$quantum groups via $\Delta$-Hall algebras}, url = {http://arxiv.org/abs/2209.00205}, year = 2022 }