*(English)*Zbl 0894.11052

The main purpose of this paper, based on a series of lectures given at Japanese Universities, is a survey of dilogarithm identities and related topics. It contains a wealth of information and an extensive list of references. The contents are conveniently described in the author’s summary.

We study the dilogarithm identities from algebraic, analytic, asymptotic, $K$-theoretic, combinatorial and representation-theoretic points of view. We prove that a lot of dilogarithm identities (hypothetically all!) can be obtained by using the five-term relation only. Among those the Coxeter, Lewin, Loxton and Browkin ones are contained. Accessibility of Lewin’s one variable and Ray’s multivariable (here for $n\le 2$ only) functional equations is given. For odd levels the ${\widehat{sl}}_{2}$ case of Kuniba-Nakanishi’s dilogarithm conjecture is proven and additional results about remainder term are obtained. The connections between dilogarithm identities and Rogers-Ramanujan-Andrews-Gordon type partition identities via their asymptotic behavior are discussed. Some new results about the string functions for level $k$ vacuum representation of the affine Lie algebra ${\widehat{sl}}_{n}$ are obtained. Connection between dilogarithm identities and algebraic $K$-theory (torsion in ${K}_{3}\left(\mathbb{R}\right))$ is discussed. Relations between crystal basis, branching functions ${b}_{\lambda}^{k{{\Lambda}}_{0}}\left(q\right)$ and Kostka-Foulkes polynomials (Lusztig’s $q$-analog of weight multiplicity) are considered. The Melzer and Milne conjectures are proven. In some special cases we are proving that the branching functions ${b}_{\lambda}^{k{{\Lambda}}_{0}}\left(q\right)$ are equal to an appropriate limit of Kostka polynomials (the so-called Thermodynamic Bethe Ansatz limit). The connection between the “finite-dimensional part of crystal base” and the Robinson-Schensted-Knuth correspondence is considered.

##### MSC:

11Z05 | Miscellaneous applications of number theory |

11-02 | Research monographs (number theory) |

33-02 | Research monographs (special functions) |

11G55 | Polylogarithms and relations with $K$-theory |

33B15 | Gamma, beta and polygamma functions |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

81T40 | Two-dimensional field theories, conformal field theories, etc. |

17B68 | Virasoro and related algebras |

11P81 | Elementary theory of partitions |

05E15 | Combinatorial aspects of groups and algebras |

17B37 | Quantum groups and related deformations |

19M05 | Miscellaneous applications of $K$-theory |