Biderivations of Null-Filiform and Filiform Zinbiel Algebras
Abstract
A biderivation of an algebra is a bilinear map which is a derivation in each argument. We describe all biderivations of finite-dimensional complex null-filiform and filiform Zinbiel algebras. For the n-dimensional null-filiform algebra, every biderivation is determined by an arbitrary value at its generating pair; the space of biderivations has dimension n and consists entirely of symmetric maps. For the three standard filiform families, we give explicit bases and determine their symmetric and skew-symmetric parts. We also treat all filiform algebras of dimensions at most four, including the exceptional families not covered by the classification in dimensions at least five. A three-dimensional algebra admits a three-dimensional space of skew-symmetric biderivations, two of which have values outside its annihilator. Up to isomorphism, it is the unique complex filiform Zinbiel algebra with this property. The proofs use generators, annihilator-valued maps, a truncated polynomial model, and conformal endomorphisms of bilinear forms.
Related
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BibTeX
@misc{yusupov2026biderivations,
title = {{Biderivations of Null-Filiform and Filiform Zinbiel Algebras}},
author = {Arislanbay Abdireymov. Bakhtiyor Yusupov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32550},
eprint = {2610.32550},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32550},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Biderivations of Null-Filiform and Filiform Zinbiel Algebras
AU - Arislanbay Abdireymov. Bakhtiyor Yusupov
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32550
UR - https://arxiv.org/abs/2610.32550
AB - A biderivation of an algebra is a bilinear map which is a derivation in each argument. We describe all biderivations of finite-dimensional complex null-filiform and filiform Zinbiel algebras. For the n-dimensional null-filiform algebra, every biderivation is determined by an arbitrary value at its generating pair; the space of biderivations has dimension n and consists entirely of symmetric maps. For the three standard filiform families, we give explicit bases and determine their symmetric and skew-symmetric parts. We also treat all filiform algebras of dimensions at most four, including the exceptional families not covered by the classification in dimensions at least five. A three-dimensional algebra admits a three-dimensional space of skew-symmetric biderivations, two of which have values outside its annihilator. Up to isomorphism, it is the unique complex filiform Zinbiel algebra with this property. The proofs use generators, annihilator-valued maps, a truncated polynomial model, and conformal endomorphisms of bilinear forms.
ER -