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In this note we consider the algebra $U_q(\hat{sl}_\infty)$ and we study thecategory O of its integrable representations. The main motivations areapplications to quantum toroidal algebras, more precisely predictions ofcharacter formulae for representations of quantum toroidal algebras. In thiscontext, we state a general positivity conjecture for representations of$U_q(\hat{sl}_\infty)$ viewed as representations of quantum toroidal algebras,that we prove for Kirillov-Reshetikhin modules.

The algebra $U_q(\hat{sl}_\infty)$ and applications