On Completeness of n-ary Hom-Nambu and Hom-Lie Superalgebras
Abstract
We introduce and study completeness for multiplicative n-ary Hom-Nambu superalgebras. Because an n-ary Hom-Nambu bracket is not necessarily totally super-skew-symmetric, we define its center as the intersection of its positional centers. A multiplicative n-ary Hom-Nambu superalgebra is complete when its center is trivial, and every αk+1-derivation is inner for all k ≥ 0. We show that this notion reduces to the usual completeness for multiplicative n-Hom-Lie superalgebras. Furthermore, we establish a completeness criterion for surjective brackets with a zero twisting map and provide a complete n-ary Hom-Nambu superalgebra that is not an n-Hom-Lie superalgebra. We also study the direct sums of complete n-Hom-Lie superalgebras and the behavior of centers under twisting. Finally, starting from a multiplicative Hom-Lie superalgebra, we consider recursively induced multiplicative n-ary Hom-Nambu superalgebras. When the twisting map is surjective, we prove that triviality of the center is preserved and reflected in this construction. We also observe that every binary αk-derivation satisfies the corresponding relative αk-derivation identity for the induced bracket. Finally, we determine the complete members in the selected low-dimensional Hom-Lie and $3$-Hom-Lie superalgebra classifications.
Related
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BibTeX
@misc{farhangdoost2026completeness,
title = {{On Completeness of $n$-ary Hom-Nambu and Hom-Lie Superalgebras}},
author = {Mohammad Reza Farhangdoost and Mohammad Reza Hafezi and Sergei Silvestrov},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30182},
eprint = {2608.30182},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30182},
}
RIS
TY - UNPB
TI - On Completeness of n-ary Hom-Nambu and Hom-Lie Superalgebras
AU - Mohammad Reza Farhangdoost
AU - Mohammad Reza Hafezi
AU - Sergei Silvestrov
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30182
UR - https://arxiv.org/abs/2608.30182
AB - We introduce and study completeness for multiplicative n-ary Hom-Nambu superalgebras. Because an n-ary Hom-Nambu bracket is not necessarily totally super-skew-symmetric, we define its center as the intersection of its positional centers. A multiplicative n-ary Hom-Nambu superalgebra is complete when its center is trivial, and every α^k+1-derivation is inner for all k ≥ 0. We show that this notion reduces to the usual completeness for multiplicative n-Hom-Lie superalgebras. Furthermore, we establish a completeness criterion for surjective brackets with a zero twisting map and provide a complete n-ary Hom-Nambu superalgebra that is not an n-Hom-Lie superalgebra. We also study the direct sums of complete n-Hom-Lie superalgebras and the behavior of centers under twisting. Finally, starting from a multiplicative Hom-Lie superalgebra, we consider recursively induced multiplicative n-ary Hom-Nambu superalgebras. When the twisting map is surjective, we prove that triviality of the center is preserved and reflected in this construction. We also observe that every binary α^k-derivation satisfies the corresponding relative α^k-derivation identity for the induced bracket. Finally, we determine the complete members in the selected low-dimensional Hom-Lie and $3$-Hom-Lie superalgebra classifications.
ER -