We give a mould theoretic reformulation of Kashiwava-Vergne Lie algebra. Based on this, we show that the bigraded version of the algebra contains Goncharov's dihedral Lie algebra. Contents 0. Introduction 1 1. Preparation on mould theory 3 1.1. Moulds and alternality 3 1.2. Ari-bracket 5 1.3. Push-invariance and pus-neutrality 6 2. Kashiwara-Vergne Lie algebra 11 2.1. Review on Kashiwara-Vergne graded Lie algebra 11 2.2. Mould theoretic reformulation 12 2.3. Kashiwara-Vergne bigraded Lie algebra 16 2.4. Mould theoretic reformulation 17 3. Dihedral Lie algebra 19 3.1. Review on the dihedral bigraded Lie algebra 20 3.2. Embedding to Kashiwara-Vergne bigraded Lie algebra 21 Appendix A. Ecalle's senary relation 25 References 28

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It is known that the special values of multiple zeta functions at non-positive arguments are indeterminate in most cases due to the occurrences of infinitely many singularities. In order to give a suitable rigorous meaning of the special values there, Furusho, Komori, Matsumoto and Tsumura introduced the desingularized values by the desingularization method to resolve all singularities. While, Ebrahimi-Fard, Manchon and Singer introduced the renormalized values to keep the "shuffle" relation by the renormalization procedure à la Connes and Kreimer. In this paper, we reveal an equivalence, that is, an explicit interrelationship between these two values. As a corollary, we also obtain an explicit formula to describe renormalized values in terms of Bernoulli numbers.

We treat desingularized multiple zeta-functions introduced by Furusho, Komori, Matsumoto and Tsumura. In this paper, we prove functional relations, which are shuffle type product formulae, between desingularized multiple zeta-functions and desingularized values.

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