Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra

@misc{ren2026brunnian,
  title = {{Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra}},
  author = {Muze Ren},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.20844},
  eprint = {2605.20844},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.20844},
}

TY  - UNPB
TI  - Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra
AU  - Muze Ren
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2605.20844
UR  - https://arxiv.org/abs/2605.20844
AB  - Schneps \cite{Schneps2012,Schneps2025} and Enriquez-Furusho \cite{EF4} proved that the double shuffle Lie algebra dmr_0 embeds into the Kashiwara--Vergne Lie algebra krv_2. We give a Brunnian braid interpretation of a related embedding into the symmetric Kashiwara--Vergne Lie algebra krv_2^sym. More precisely, the map φlongmapsto (φ(−x_0−x_1,x_0),φ(−x_0−x_1,x_1)) defines an injective Lie algebra homomorphism from the subalgebra of dmr_0 satisfying the condition [x_0,φ(−x_0−x_1,x_0)] +[x_1,φ(−x_0−x_1,x_1)]=0 into krv_2^sym. The proof reformulate the double shuffle and symmetric Kashiwara--Vergne relations through abelianizations of Brunnian Lie algebras associated with the disk and punctured disks. We generalize this inclusion in two directions. First, replacing these abelianizations by higher lower central series quotients yields generalizations of relations and implications among them. Second, we establish explicit identities relating the linear pentagon defect to the stuffle coproduct, the divergence map, and the necklace cobracket.
ER  -