Dilogarithm Identities

We study the dilogarithm identities from algebraic, analytic, asymptotic,$K$-theoretic, combinatorial and representation-theoretic points of view. Weprove that a lot of dilogarithm identities (hypothetically all !) can beobtained by using the five-term relation only. Among those the Coxeter, Lewin,Loxton and Browkin ones are contained. Accessibility of Lewin's one variableand Ray's multivariable (here for $n\le 2$ only) functional equations is given.For odd levels the $\hat{sl_2}$ case of Kuniba-Nakanishi's dilogarithmconjecture is proven and additional results about remainder term are obtained.The connections between dilogarithm identities andRogers-Ramanujan-Andrews-Gordon type partition identities via their asymptoticbehavior are discussed. Some new results about the string functions for level$k$ vacuum representation of the affine Lie algebra $\hat{sl_n}$ are obtained.Connection between dilogarithm identities and algebraic $K$-theory (torsion in$K_3({\bf R})$) is discussed. Relations between crystal basis, branchingfunctions $b_{\lambda}^{k\Lambda_0}(q)$ and Kostka-Foulkes polynomials(Lusztig's $q$-analog of weight multiplicity) are considered. The Melzer andMilne conjectures are proven. In some special cases we are proving that thebranching functions $b_{\lambda}^{k\Lambda_0}(q)$ are equal to an appropriatelimit of Kostka polynomials (the so-called Thermodynamic Bethe Ansatz limit).Connection between "finite-dimensional part of crystal base" andRobinson-Schensted-Knuth correspondence is considered.Comment: 96 page