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## Homomorphisms between different quantum toroidal and affine Yangian algebras

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of sl*n*, denoted by U(*n*)*q*1,*q*2,*q*3 and Y(*n*)*h*1,*h*2,*h*3, respectively. Our motivation arises from the milestone work of Gautam and Toledano Laredo, where a similar relation between the quantum loop algebra *U**q*(*L*g) and the Yangian *Y**h*(g) has been established by constructing an isomorphism of C[[ℏ]]-algebras Φ:*U*ˆexp(ℏ)(*L*g)→*Y*ˆℏ(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 *q**i*=exp(ℏ*i*) for *i*=1,2,3.

In the current paper, we are interested in the more general relation: q1=*ω**m**n**e**h*1/*m*,q2=*e**h*2/*m*,q3=*ω*−1*m**n**e**h*3/*m*, where *m*,*n*∈N and *ω**m**n* is an *m**n*-th root of 1. Assuming *ω**m**m**n* is a primitive *n*-th root of unity, we construct a homomorphism Φ*ω**m**n**m*,*n* from the completion of the formal version of U(*m*)q1,q2,q3 to the completion of the formal version of Y(*m**n*)*h*1/*m**n*,*h*2/*m**n*,*h*3/*m**n*. We propose two proofs of this result: (1) by constructing the compatible isomorphism between the faithful representations of the algebras; (2) by combining the direct verification of Gautam and Toledano Laredo for the classical setting with the shuffle approach.