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

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

2009-03-27

arxiv.org/abs/0903.4828v2

Mathematics

Representation Theory

Some references addes and typos corrected

Scientific paper

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

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