On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra

2017/06/26 by Furusho, Hidekazu, Raphael, Elise, Komiyama, Nao +1 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1706.08299

openalex publication_date 2017/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The goal of this article is to define a linearized or depth-graded version \mathfraklkv, and a closely related elliptic version \mathfrakkrv_ell, of the Kashiwara-Vergne Lie algebra \mathfrakkrv originally constructed by Alekseev and Torossian as the space of solutions to the linearized Kashiwara-Vergne problem. We show how the elliptic Lie algebra \mathfrakkrv_ell is related to earlier constructions of elliptic versions \mathfrakgrt_ell and \mathfrakds_ell of the Grothendieck-Teichmüller Lie algebra \mathfrakgrt and the double shuffle Lie algebra \mathfrakds. In particular we show that there is an injective Lie morphism \mathfrakds_ell\hookrightarrow \mathfrakkrv_ell, and an injective Lie algebra morphism \mathfrakkrv→ \mathfrakkrv_ell extending the known morphisms \mathfrakgrt\hookrightarrow\mathfrakgrt_ell (Enriquez section) and \mathfrakds→\mathfrakds_ell (Écalle map).

Cited by

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

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