## Abstract

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of sl_{n}, denoted by U_{q1,q2,q3 } ^{(n)} and Y_{h1,h2,h3 } ^{(n)}, respectively. Our motivation arises from the milestone work [11], where a similar relation between the quantum loop algebra U_{q}(Lg) and the Yangian Y_{h}(g) has been established by constructing an isomorphism of C[[ħ]]-algebras Φ:Uˆ_{exp(ħ)}(Lg)⟶∼Yˆ_{ħ}(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 (see [11,22]). In the current paper, we are interested in the more general relation: q_{1}=ω_{mn}e^{h1/m},q_{2}=e^{h2/m},q_{3}=ω_{mn} ^{−1}e^{h3/m}, where m,n≥1 and ω_{mn} is an mn-th root of 1. Assuming ω_{mn} ^{m} is a primitive n-th root of unity, we construct a homomorphism Φ_{m,n} ^{ωmn } between the completions of the formal versions of U_{q1,q2,q3 } ^{(m)} and Y_{h1/mn,h2/mn,h3/mn} ^{(mn)}.

Original language | English |
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Pages (from-to) | 867-899 |

Number of pages | 33 |

Journal | Journal of Pure and Applied Algebra |

Volume | 223 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2019 |