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.

arXiv comment55 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

References in corpus (2)

• Convex Bases of PBW type for Quantum Affine Algebras
• The trigonometric Casimir connection of a simple Lie algebra

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