Yangians and quantum loop algebras
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.
Keywords
Cite
@article{arxiv.1012.3687,
title = {Yangians and quantum loop algebras},
author = {Sachin Gautam and Valerio Toledano-Laredo},
journal= {arXiv preprint arXiv:1012.3687},
year = {2013}
}
Comments
55 pages. Introduction expanded, exposition in section 4 improved, isomorphism between completions for gl(n) added in section 7, a few references added, minor changes throughout. Final version. Published online in Selecta Mathematica