Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 104, 5 pages      arXiv:1210.0226      https://doi.org/10.3842/SIGMA.2012.104

Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras

Tomoki Nakanishi
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8604, Japan

Received November 15, 2012, in final form December 22, 2012; Published online December 25, 2012

Abstract
Recently Cherednik and Feigin [arXiv:1209.1978] obtained several Rogers-Ramanujan type identities via the nilpotent double affine Hecke algebras (Nil-DAHA). These identities further led to a series of dilogarithm identities, some of which are known, while some are left conjectural. We confirm and explain all of them by showing the connection with Y-systems associated with (untwisted and twisted) quantum affine Kac-Moody algebras.

Key words: double affine Hecke algebra; dilogarithm; Y-system.

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References

  1. Cecotti S., Neitzke A., Vafa C., R-twisting and 4d/2d correspondences, arXiv:1006.3435.
  2. Chapoton F., Functional identities for the Rogers dilogarithm associated to cluster Y-systems, Bull. London Math. Soc. 37 (2005), 755-760.
  3. Cherednik I., Feigin B., Rogers-Ramanujan type identities and Nil-DAHA, arXiv:1209.1978 (especially version 2).
  4. Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T- and Y-systems, dilogarithm identities, and cluster algebras I: Type Br, Publ. Res. Inst. Math. Sci., to appear, arXiv:1001.1880.
  5. Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T- and Y-systems, dilogarithm identities, and cluster algebras II: Types Cr, F4, and G2, Publ. Res. Inst. Math. Sci., to appear, arXiv:1001.1881.
  6. Inoue R., Iyama O., Kuniba A., Nakanishi T., Suzuki J., Periodicities of T-systems and Y-systems, Nagoya Math. J. 197 (2010), 59-174, arXiv:0812.0667.
  7. Kashaev R.M., Nakanishi T., Classical and quantum dilogarithm identities, SIGMA 7 (2011), 102, 29 pages, arXiv:1104.4630.
  8. Lee C.H., Nahm's conjecture and Y-system, arXiv:1109.3667.
  9. Nakanishi T., Dilogarithm identities for conformal field theories and cluster algebras: simply laced case, Nagoya Math. J. 202 (2011), 23-43, arXiv:0909.5480.

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