T-systems, Y-systems, and cluster algebras: Tamely laced case

The T-systems and Y-systems are classes of algebraic relations originally associated with quantum affine algebras and Yangians. Recently they were generalized to quantum affinizations of quantum Kac-Moody algebras associated with a wide class of generalized Cartan matrices which we say tamely laced. Furthermore, in the simply laced case, and also in the nonsimply laced case of finite type, they were identified with relations arising from cluster algebras. In this note we generalize such an identification to any tamely laced Cartan matrices, especially to the nonsimply laced ones of nonfinite type.Comment: 31 pages, final version to appear in the festschrift volume for Tetsuji Miwa, "Infinite Analysis 09: New Trends in Quantum Integrable Systems

Structure of seeds in generalized cluster algebras

We study generalized cluster algebras introduced by Chekhov and Shapiro. When the coefficients satisfy the normalization and quasi-reciprocity conditions, one can naturally extend the structure theory of seeds in the ordinary cluster algebras by Fomin and Zelevinsky to generalized cluster algebras. As the main result, we obtain formulas expressing cluster variables and coefficients in terms of c-vectors, g-vectors, and F-polynomials.Comment: 15 pages; v2:minor revision; v3:typos corrected, to appear in Pacific J. Mat

Note on dilogarithm identities from nilpotent double affine Hecke algebras

Recently Cherednik and Feigin 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.Comment: 5 pages; v2: Section 4 added, journal version in SIGM

Spectra in Conformal Field Theories from the Rogers Dilogarithm

We propose a system of functional relations having a universal form connected to the $U_q(X^{(1)}_r)$ Bethe ansatz equation. Based on the analysis of it, we conjecture a new sum formula for the Rogers dilogarithm function in terms of the scaling dimensions of the $X^{(1)}_r$ parafermion conformal field theory.Comment: 10 pages, MRR-009-92, SMS-042-9