Two descriptions of the quantum affine algebra U v(sl̂ 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 Q → and the category of coherent sheaves on ℙ 1. Using this approach, we show that the Drinfeld-Beck isomorphism for the quantized enveloping algebra U v(sl̂ 2) is a corollary of an equivalence between the derived categories D b(Rep(Q →)) and D b(Coh(ℙ 1)). This technique allows to reprove several results on the integral form of U v(sl̂ 2). © 2012 Glasgow Mathematical Journal Trust.