Crystal Bases, Dilogarithm Identities and Torsion in Algebraic K–groups Edward Frenkel and András Szenes

and δj = sin 2 π k+2 / sin π(j+1) k+2 . They were first proved by Kirillov and Reshetikhin [1]. The right hand side of this identity is equal to π 2 6 times the central charge of an integrable level k representation of the affine Kac-Moody Lie algebra ŝl2. This number appears in the asymptotics of the character of such a representation. Following the general approach taken in our previous work [2], we will obtain the left hand side as a result of an alternative calculation of this asymptotics. We will use a combinatorial description of the so-called crystal basis [6] of an integrable ŝl2-module V k of level k, found by Jimbo, Misra, Miwa and Okado in [3] (cf. also [4, 5]). They defined a weight function ω on the crystal basis vectors, such that if we sum up q over all of them, we obtain the character of V . We will introduce an increasing system of finite subsets of the set of all crystal basis vectors and show that the corresponding partial characters are related by a q-recurrence relation. This will give us a formula for the character of V k via the product of infinitely many matrices. In the limit when q → 1, we will obtain a formula for the asymptotics as an integral, which is then shown to be equal to the left hand side of (1.1). The dilogarithm identities are closely connected with elements of torsion in the group K3(F ), where F is a totally real field of algebraic numbers. We construct such