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Infinitesimal Dilogarithm Satisfies Cluster Identities

Sinan Unver

TL;DR

This work shows that infinitesimal dilogarithms and Kontsevich's $1\\frac{1}{2}$-logarithm satisfy the same cluster-period identities as those arising from periods in cluster patterns, and it proves that in the infinitesimal setting these cluster identities are consequences of the pentagon relation. By situating $\\ell i_{m,w}$ within the Bloch complex and establishing an infinitesimal reduction theorem, the authors connect cluster identities to fundamental regulator constructions in algebraic K-theory. The characteristic-$p$ case, via $\\ell i^{(p)}_{2}$, yields a Kontsevich-type $4$-term functional equation for the $1\\frac{1}{2}$-logarithm and a corresponding $B_{2}$-type relation, illustrating how these infinitesimal objects capture cluster-period relations across characteristics. The results unify cluster algebras, dilogarithm identities, and regulator theory in a precise infinitesimal framework with explicit formulas and decompositions by weight components.

Abstract

In this paper, we show that the infinitesimal dilogarithm and Kontsevich's one-and-a-half logarithm function satisfies the identities which result from periods in cluster patterns. We also prove that these cluster identities are a consequence of the pentagon relation in the infinitesimal case.

Infinitesimal Dilogarithm Satisfies Cluster Identities

TL;DR

This work shows that infinitesimal dilogarithms and Kontsevich's -logarithm satisfy the same cluster-period identities as those arising from periods in cluster patterns, and it proves that in the infinitesimal setting these cluster identities are consequences of the pentagon relation. By situating within the Bloch complex and establishing an infinitesimal reduction theorem, the authors connect cluster identities to fundamental regulator constructions in algebraic K-theory. The characteristic- case, via , yields a Kontsevich-type -term functional equation for the -logarithm and a corresponding -type relation, illustrating how these infinitesimal objects capture cluster-period relations across characteristics. The results unify cluster algebras, dilogarithm identities, and regulator theory in a precise infinitesimal framework with explicit formulas and decompositions by weight components.

Abstract

In this paper, we show that the infinitesimal dilogarithm and Kontsevich's one-and-a-half logarithm function satisfies the identities which result from periods in cluster patterns. We also prove that these cluster identities are a consequence of the pentagon relation in the infinitesimal case.

Paper Structure

This paper contains 6 sections, 4 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Infinitesimal cluster relations
  3. Inifinitesimal dilogarithm
  4. Cluster identities for the infinitesimal dilogarithm.
  5. Inifinitesimal reduction problem
  6. Cluster relations for Kontsevich $1\frac{1}{2}$-logarithm

Key Result

Lemma 2.1.2

Suppose that we are given a $\nu$-periodic sequence of mutations in a cluster pattern as in (periodic-sequence). Let $k$ be a field with ${\rm char}(k)\neq$2. There is a proper algebraic set $X \subseteq \mathbb{A}^{n}_{k}$ inside the $n$-dimensional affine space $\mathbb{A}^{n}_{k}$ over $k$ such t

Theorems & Definitions (10)

  • Example 2.1.1
  • Lemma 2.1.2
  • proof
  • Theorem 2.1.3
  • proof
  • Theorem 2.1.4
  • proof
  • Theorem 2.2.1
  • proof
  • Example 2.2.2