A characterization of central extensions relative to n-nilpotent objects
Abstract
We characterize the central extensions, in the sense of the categorical theory of central extensions of Janelidze and Kelly, relative to the Birkhoff subcategory of n-nilpotent objects of a semi-abelian and algebraically coherent category. An extension f : A → B with kernel K turns out to be central precisely when the iterated Huq commutator [A,[A,…,[A,K]…]], with n occurrences of A, vanishes. For n=2 this condition reads [A,[K,A]] = 0, that is, [K,A] is a central subobject of A. In the special case of groups this description was obtained by Everaert and the first author. The present proof is categorical, and applies in particular to Lie algebras, to crossed modules, to cocommutative Hopf algebras over a field, and to any category of interest.
Related
- Counting Nilpotent Galois extensions with prescribed genus number
- Higher Cotorsion Pairs on Image-Nilpotent η-Extensions
- Nilpotent Jacobian maps in dimension three and stable tameness in block extensions
- On the n-perfectness under split nilpotent extensions
- Central Test Set and Central Test Rank of Free Nilpotent Lie Algebras of Rank 2
BibTeX
@misc{gran2026characterization,
title = {{A characterization of central extensions relative to n-nilpotent objects}},
author = {Marino Gran and James R. A. Gray},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17389},
eprint = {2610.17389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17389},
}
RIS
TY - UNPB
TI - A characterization of central extensions relative to n-nilpotent objects
AU - Marino Gran
AU - James R. A. Gray
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17389
UR - https://arxiv.org/abs/2610.17389
AB - We characterize the central extensions, in the sense of the categorical theory of central extensions of Janelidze and Kelly, relative to the Birkhoff subcategory of n-nilpotent objects of a semi-abelian and algebraically coherent category. An extension f : A → B with kernel K turns out to be central precisely when the iterated Huq commutator [A,[A,…,[A,K]…]], with n occurrences of A, vanishes. For n=2 this condition reads [A,[K,A]] = 0, that is, [K,A] is a central subobject of A. In the special case of groups this description was obtained by Everaert and the first author. The present proof is categorical, and applies in particular to Lie algebras, to crossed modules, to cocommutative Hopf algebras over a field, and to any category of interest.
ER -