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HIGHER LEVEL FOCK SPACES AND AFFINE YANGIAN

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Abstract

We construct actions of the affine Yangian of type A on higher level Fock spaces by extending known actions of the Yangian of finite type A due to Uglov. This is a degenerate analog of a result by Takemura–Uglov, which constructed actions of the quantum toroidal algebra on higher level q-deformed Fock spaces.

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References

  1. В. Г. Дринфельд, Вырожденные аффинные алгебры Гекке и янгианы, Функц. анализ и его прил. 20 (1986), вьш. 1, 69–70. Engl. transl.: V. Drinfel’d, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Its Appl. 20 (1986), no. 1, 58–60.

  2. B. Feigin, M. Finkelberg, A. Negut, L. Rybnikov, Yangians and cohomology rings of Laumon spaces, Selecta Math. (N.S.) 17 (2011), no. 3, 573–607.

    Article  MathSciNet  Google Scholar 

  3. N. Guay, Cherednik algebras and Yangians, Int. Math. Res. Not. (2005), no. 57, 3551–3593.

  4. N. Guay, Affine Yangians and deformed double current algebras in type A, Adv. Math. 211 (2007), no. 2, 436–484.

    Article  MathSciNet  Google Scholar 

  5. R. Kodera, Affine Yangian action on the Fock space, arXiv:1506.01246 (2015).

  6. B. Leclerc, Fock space representations of \( {U}_q\left({\widehat{\mathfrak{sl}}}_n\right) \), in: Geometric Methods in Re-presentation Theory, I, Sémin. Congr., Vol. 24, Soc. Math. France, Paris, 2012, pp. 343–385.

  7. Y. Saito, K. Takemura, D. Uglov, Toroidal actions on level 1 modules of \( {U}_q\left({\widehat{\mathfrak{sl}}}_n\right) \), Transform. Groups 3 (1998), no. 1, 75–102.

    Article  MathSciNet  Google Scholar 

  8. K. Takemura, D. Uglov, Level-0 action of \( {U}_q\left({\widehat{\mathfrak{sl}}}_n\right) \) on the q-deformed Fock spaces, Comm. Math. Phys. 190 (1998), no. 3, 549–583.

    Article  MathSciNet  Google Scholar 

  9. K. Takemura, D. Uglov, Representations of the quantum toroidal algebra on highest weight modules of the quantum affine algebra of type \( {\mathfrak{gl}}_N \), Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, 407–450.

    Article  MathSciNet  Google Scholar 

  10. D. Uglov, Symmetric functions and the Yangian decomposition of the Fock and basic modules of the affine Lie algebra \( {\widehat{\mathfrak{sl}}}_N \), in: Quantum Many-body Problems and Representation Theory, MSJ Mem., Vol. 1, Math. Soc. Japan, Tokyo, 1998, pp. 183–241.

  11. D. Uglov, Yangian actions on higher level irreducible integrable modules of \( {\widehat{\mathfrak{gl}}}_n \), arXiv:9802048 (1998).

  12. D. Uglov, Canonical bases of higher-level q-deformed Fock spaces and Kazhdan–Lusztig polynomials, in: Physical Combinatorics (Kyoto, 1999), Progr. Math., Vol. 191, Birkhäuser Boston, Boston, MA, 2000, pp. 249–299.

  13. M. Varagnolo, Quiver varieties and Yangians, Lett. Math. Phys. 53 (2000), no. 4, 273–283.

    Article  MathSciNet  Google Scholar 

  14. M. Varagnolo, E. Vasserot, Double-loop algebras and the Fock space, Invent. Math. 133 (1998), no. 1, 133–159.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe, 657-8501, Japan

    RYOSUKE KODERA

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  1. RYOSUKE KODERA
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KODERA, R. HIGHER LEVEL FOCK SPACES AND AFFINE YANGIAN. Transformation Groups 23, 939–962 (2018). https://doi.org/10.1007/s00031-018-9491-8

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