Yangians and quantum loop algebras

TL;DR

This paper explicitly constructs an algebra homomorphism linking quantum loop algebras and Yangians for semisimple Lie algebras, demonstrating their structural relationship and geometric action compatibility.

Contribution

It provides an explicit algebra homomorphism from quantum loop algebras to Yangians and shows it becomes an isomorphism upon completion, extending understanding of their connection.

Findings

Constructed an explicit algebra homomorphism Phi over Q[[h]]

Proved Phi is an isomorphism after completion with respect to the evaluation ideal

Intertwines geometric actions on equivariant K-theory and cohomology for g=gl_n

Abstract

Let g be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra U_h(Lg) of g degenerates to the Yangian Y_h(g). We strengthen this result by constructing an explicit algebra homomorphism Phi defined over Q[[h]] from U_h(Lg) to the completion of Y_h(g) with respect to its grading. We show moreover that Phi becomes an isomorphism when the quantum loop algebra is completed with respect to its its evaluation ideal. We construct a similar homomorphism for g=gl_n and show that it intertwines the geometric actions of U_h(L gl_n) and Y(gl_n) on the equivariant K-theory and cohomology of the variety of n-step flags in C^d constructed by Ginzburg and Vasserot.