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Anchor Radicals and Metric Rigidity in n-Lie--Rinehart Algebras
Abstract
For an n-Lie--Rinehart algebra, n>2, the anchor is defined on Aextn−1L, so its kernel does not determine a distinguished submodule of L. We introduce the anchor radical Rad(ρ)⊂ L and prove that it is an adjoint-stable ideal on which the bracket is A-multilinear. We then derive an explicit obstruction to the A-linearity of metric compatibility in fundamental directions. This obstruction vanishes on Rad(ρ), whereas a strongly nondegenerate compatible metric on all of L forces the anchor to vanish. These results lead to a natural notion of orthogonal structure on the anchor radical. For anchors induced by Nambu tensors, the radical is identified with the cotangent annihilator of the tensor.
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BibTeX
@misc{nsiloulou2026anchor,
title = {{Anchor Radicals and Metric Rigidity in $n$-Lie--Rinehart Algebras}},
author = {Irmely Gladesh Mabanza Nsiloulou and Basile Guy Richard Bossoto},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31538},
eprint = {2609.31538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31538},
}
RIS
TY - UNPB
TI - Anchor Radicals and Metric Rigidity in n-Lie--Rinehart Algebras
AU - Irmely Gladesh Mabanza Nsiloulou
AU - Basile Guy Richard Bossoto
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31538
UR - https://arxiv.org/abs/2609.31538
AB - For an n-Lie--Rinehart algebra, n>2, the anchor is defined on Aextn−1L, so its kernel does not determine a distinguished submodule of L. We introduce the anchor radical Rad(ρ)⊂ L and prove that it is an adjoint-stable ideal on which the bracket is A-multilinear. We then derive an explicit obstruction to the A-linearity of metric compatibility in fundamental directions. This obstruction vanishes on Rad(ρ), whereas a strongly nondegenerate compatible metric on all of L forces the anchor to vanish. These results lead to a natural notion of orthogonal structure on the anchor radical. For anchors induced by Nambu tensors, the radical is identified with the cotangent annihilator of the tensor.
ER -