Homomorphisms between different quantum toroidal and affine Yangian algebras

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of sln, denoted by Uq1,q2,q3(n) and Yh1,h2,h3(n), respectively. Our motivation arises from the milestone work [11], where a similar relation between the quantum loop algebra Uq(Lg) and the Yangian Yh(g) has been established by constructing an isomorphism of C[[ħ]]-algebras Φ:Uˆexp⁡(ħ)(Lg)⟶∼Yˆħ(g) (with ˆ standing for the appropriate completions). These two completions model the behavior of the algebras in the formal neighborhood of h=0. The same construction can be applied to the toroidal setting with qi=exp⁡(ħi) for i=1,2,3 (see [11], [22]). In the current paper, we are interested in the more general relation: q1=ωmneh1/m,q2=eh2/m,q3=ωmn−1eh3/m, where m,n≥1 and ωmn is an mn-th root of 1. Assuming ωmnm is a primitive n-th root of unity, we construct a homomorphism Φm,nωmn between the completions of the formal versions of Uq1,q2,q3(m) and Yh1/mn,h2/mn,h3/mn(mn).