Two descriptions of the quantum affine algebra \(U_{v}(\widehat{\mathfrak{sl}}_2)\) via Hall algebra approach (Q2882493)

In this paper the authors study the relation between the composition algebra of the category \(\mathrm{Rep}(\overrightarrow{Q})\) of representations of the Kronecker quiver and the composition algebra of the category \(\mathrm{Coh}(\mathbb{P}^1)\) of coherent sheaves on the projective line. By comparing the reduced Drinfeld doubles the authors show that the Drinfeld-Beck isomorphism for the quantized enveloping algebra \(U_v(\hat{\mathfrak{sl}}_2)\) is a corollary of an equivalence between the corresponding derived categories: \(D^b(\mathrm{Rep}({\overrightarrow{Q}}))\cong D^b(\mathrm{Coh}(\mathbb{P}^1))\). Using the fact that such an equivalence commutes with Serre functors, the authors derive that the Drinfeld-Beck isomorphism commutes with the Coxeter transformation. As an application, the authors reprove several known technical statements on the integral form of \(U_v(\hat{\mathfrak{sl}}_2)\).