Two descriptions of the quantum affine algebra $U_v(\hat{\mathfrak{sl}}_2)$ via Hall algebra approach

TL;DR

This paper compares two algebraic descriptions of the quantum affine algebra $U_v(\\hat{\rak{sl}}_2)$ using Hall algebra techniques, revealing their equivalence through derived category isomorphisms and rederiving key results.

Contribution

It establishes a new connection between the Hall algebra approach and the Drinfeld--Beck isomorphism via derived category equivalences.

Findings

Proves the Drinfeld--Beck isomorphism follows from derived category equivalence.

Reproduces key results on the integral form of $U_v(\hat{\frak{sl}}_2)$.

Shows the equivalence of categories of representations and coherent sheaves.

Abstract

We compare the reduced Drinfeld doubles of the composition subalgebras of the category of representations of the Kronecker quiver $\overr{Q}$ and of the category of coherent sheaves on ${\mathbb P}^1$. Using this approach, we 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 derived categories $D^b\bigl(\Rep(\overr{Q})\bigr)$ and $D^b\bigl(\Coh({\mathbb P}^1)\bigr)$. This technique also allows to reprove several technical results on the integral form of $U_v(\hat{\mathfrak{sl}}_2)$.