Local and 2 local 1/2-derivation of n-dimensional totally graded filiform Lie algebras
Abstract
This article provides a complete algebraic description of 1/2-derivations, local 1/2-derivations, and 2-local 1/2-derivations on n-dimensional totally graded complex filiform Lie algebras of maximum length. Based on the foundational classification framework established by Janez Bernik (2020), we systematically determine the vector spaces of 1/2-derivations for the six infinite structural sequences (m0(n), m2(n), W+(n), m0,1(n), m0,2(n), m0,3(n)) and the five exceptional one-parameter families (g7,α through g11,α). By analyzing the pointwise local evaluation equations via parametric matrix systems, we establish the structural linearity and rigidity of local 1/2-derivations. In contrast, we demonstrate that the independent parameters residing in the boundary rows of the 1/2-derivation matrices provide sufficient degrees of freedom to bypass linearity constraints. Exploiting these boundary configurations, we explicitly construct pure non-linear and non-additive 2-local 1/2-derivations leveraging the homogeneous function of degree one, f(z1, z2) = z13 / (z12 + z22), thereby defining the exact boundary where local rigidity fails.
Related
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BibTeX
@misc{arzikulov2026local,
title = {{Local and 2 local $\frac12$-derivation of $n$-dimensional totally graded filiform Lie algebras}},
author = {Farkhodzhon Arzikulov and Mirzobek Shodiev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01556},
eprint = {2610.01556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01556},
}
RIS
TY - UNPB
TI - Local and 2 local 1/2-derivation of n-dimensional totally graded filiform Lie algebras
AU - Farkhodzhon Arzikulov
AU - Mirzobek Shodiev
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01556
UR - https://arxiv.org/abs/2610.01556
AB - This article provides a complete algebraic description of 1/2-derivations, local 1/2-derivations, and 2-local 1/2-derivations on n-dimensional totally graded complex filiform Lie algebras of maximum length. Based on the foundational classification framework established by Janez Bernik (2020), we systematically determine the vector spaces of 1/2-derivations for the six infinite structural sequences (m_0(n), m_2(n), W^+(n), m_0,1(n), m_0,2(n), m_0,3(n)) and the five exceptional one-parameter families (g_7,α through g_11,α). By analyzing the pointwise local evaluation equations via parametric matrix systems, we establish the structural linearity and rigidity of local 1/2-derivations. In contrast, we demonstrate that the independent parameters residing in the boundary rows of the 1/2-derivation matrices provide sufficient degrees of freedom to bypass linearity constraints. Exploiting these boundary configurations, we explicitly construct pure non-linear and non-additive 2-local 1/2-derivations leveraging the homogeneous function of degree one, f(z_1, z_2) = z_1^3 / (z_1^2 + z_2^2), thereby defining the exact boundary where local rigidity fails.
ER -