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 Uqjavax.xml.bind.JAXBElement@4e20f6e0,qjavax.xml.bind.JAXBElement@3438b591,qjavax.xml.bind.JAXBElement@3d51e807(n) and Yhjavax.xml.bind.JAXBElement@47137645,hjavax.xml.bind.JAXBElement@ecd0fb1,hjavax.xml.bind.JAXBElement@7fa8b7c4(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=ωmnehjavax.xml.bind.JAXBElement@4d8b79e6/m,q2=ehjavax.xml.bind.JAXBElement@11768f4c/m,q3=ωmn−1ehjavax.xml.bind.JAXBElement@7720ba54/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,nωjavax.xml.bind.JAXBElement@4d0c25e3 between the completions of the formal versions of Uqjavax.xml.bind.JAXBElement@5b5e31ed,qjavax.xml.bind.JAXBElement@7ef7c925,qjavax.xml.bind.JAXBElement@cf72eb5(m) and Yhjavax.xml.bind.JAXBElement@207821f5/mn,hjavax.xml.bind.JAXBElement@a83ad4e/mn,hjavax.xml.bind.JAXBElement@4afdf3da/mn(mn).