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arXiv · 2609.31538

Anchor Radicals and Metric Rigidity in $n$-Lie--Rinehart Algebras

Irmely Gladesh Mabanza Nsiloulou·Basile Guy Richard Bossoto
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Abstract

For an $n$-Lie--Rinehart algebra, $n>2$, the anchor is defined on $\Aext{n-1}L$, so its kernel does not determine a distinguished submodule of $L$. We introduce the anchor radical $\Rad(ρ)\subset L$ 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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Irmely Gladesh Mabanza Nsiloulou, Basile Guy Richard Bossoto. 2026-09-25. Anchor Radicals and Metric Rigidity in $n$-Lie--Rinehart Algebras. https://arxiv.org/abs/2609.31538

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