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 U(n)q1,q2,q3 and Y(n)h1,h2,h3, respectively. Our motivation arises from the milestone work of Gautam and Toledano Laredo, 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.
In the current paper, we are interested in the more general relation: q1=ωmneh1/m,q2=eh2/m,q3=ω−1mneh3/m, where m,n∈N and ωmn is an mn-th root of 1. Assuming ωmmn is a primitive n-th root of unity, we construct a homomorphism Φωmnm,n from the completion of the formal version of U(m)q1,q2,q3 to the completion of the formal version of Y(mn)h1/mn,h2/mn,h3/mn. 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.