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Yangians and quantum loop algebras

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Abstract

Let \(\mathfrak{g }\) be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra \(U_\hbar (L\mathfrak g )\) of \(\mathfrak{g }\) degenerates to the Yangian \({Y_\hbar (\mathfrak g )}\). We strengthen this result by constructing an explicit algebra homomorphism \(\Phi \) from \(U_\hbar (L\mathfrak g )\) to the completion of \({Y_\hbar (\mathfrak g )}\) with respect to its grading. We show moreover that \(\Phi \) becomes an isomorphism when \({U_\hbar (L\mathfrak g )}\) is completed with respect to its evaluation ideal. We construct a similar homomorphism for \(\mathfrak{g }=\mathfrak{gl }_n\) and show that it intertwines the actions of \(U_\hbar (L\mathfrak gl _{n})\) and \(Y_\hbar (\mathfrak gl _{n})\) on the equivariant \(K\)-theory and cohomology of the variety of \(n\)-step flags in \({\mathbb{C }}^d\) constructed by Ginzburg–Vasserot.

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Notes

  1. 1.

    \( \widetilde{U}^0\) is isomorphic to the subalgebra \(U^0\) of \(U_\hbar (L\mathfrak g )\) generated by \(\{H _{i,r}\}_{i\in \mathbf{I },r\in \mathbb Z }\) by the PBW Theorem for \(U_\hbar (L\mathfrak g )\) [1], but we shall not need this fact.

  2. 2.

    note that the expansions in \(z^ {\pm 1}\) are related by the symmetry \(\vartheta _m(\zeta ^{-1})=\vartheta _{-m}(\zeta )\).

  3. 3.

    our conventions are adapted to [15, 34] They differ from those of [25] by the permutation \(e_{i,r}\leftrightarrow f_{i,r}\) and the relabelling \(\theta _{i,r}\leftrightarrow h_{i,r}\).

  4. 4.

    Note the difference between \(J(v)\) and the function \(G(v)\) used in Sect. 4.1 for constructing the solutions for simple Lie algebras: \(J(v) = G(v) + \frac{v}{2}\).

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Acknowledgments

We are very grateful to Ian Grojnowski from whom we learned that the quantum loop algebra and Yangian should be isomorphic after appropriate completions. His explanations and friendly insistence helped us overcome our initial doubts. We are also grateful to V. Drinfeld for showing us a proof that finite-dimensional representations separate elements of the Yangian and allowing us to reproduce it in Sect. 8. We would also like to thank N. Guay for sharing a preliminary version of the preprint [17] and E. Vasserot for useful discussions.

Author information

Correspondence to Valerio Toledano Laredo.

Additional information

Both authors were supported by NSF grants DMS-0707212 and DMS-0854792.

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Gautam, S., Toledano Laredo, V. Yangians and quantum loop algebras. Sel. Math. New Ser. 19, 271–336 (2013). https://doi.org/10.1007/s00029-012-0114-2

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Keywords

  • Affine quantum groups
  • Yangian
  • Quantum loop algebra

Mathematics Subject Classification (1991)

  • 17B37 (17B67 · 82B23 · 14F43)