Numerical computation of linearized KV and the Deligne-Drinfeld and Broadhurst-Kreimer conjectures

TL;DR

This paper performs numerical computations to verify several deep conjectures in Lie algebra theory and number theory, including the Kashiwara-Vergne algebra, Deligne-Drinfeld, and Broadhurst-Kreimer conjectures, up to certain weights.

Contribution

It provides the first extensive numerical verification of these conjectures up to weight 29, confirming their validity in those cases.

Findings

Confirmed the dimension conjecture of lkv in low weights.

Validated the Deligne-Drinfeld conjecture up to weight 29.

Supported the Broadhurst-Kreimer conjecture for multiple zeta values at high weights.

Abstract

We compute numerically the dimensions of the graded quotients of the linearized Kashiwara-Vergne Lie algebra lkv in low weight, confirming a conjecture of Raphael-Schneps in those weights. The Lie algebra lkv appears in a chain of inclusions of Lie algebras, including also the linearized double shuffle Lie algebra and the (depth associated graded of the) Grothendieck-Teichm\"uller Lie algebra. Hence our computations also allow us to check the validity of the Deligne-Drinfeld conjecture on the structure of the Grothendieck-Teichm\"uller group up to weight 29, and (a version of) the the Broadhurst-Kreimer conjecture on the number of multiple zeta values for a range of weight-depth pairs significantly exceeding the previous bounds. Our computations also verify a conjecture by Alekseev-Torossian on the Kashiwara-Vergne Lie algebra up to weight 29.