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

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 $U_v({\widehat{\mathfrak{sl}}}_2)$ 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 $U_v({\widehat{\mathfrak{sl}}}_2)$.

Some references addes and typos corrected

References in corpus (2)

• The composition algebra of an affine quiver
• Composition algebra of a weighted projective line

Cited by in corpus (2)

• Double Hall algebras and derived equivalences
• Composition algebra of a weighted projective line