Generalized Derivations and Canonical Envelopes of Complex α2-Weighted Ω-Hom-Lie Superalgebras
- 1Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
Abstract
We study generalized derivations of complex multiplicative α2-weighted Ω-Hom-Lie superalgebras. The chosen structural identity is shown to be stable under Yau twisting, and every regular algebra in this class untwists canonically to an Ω-Lie superalgebra. To retain the auxiliary operators that occur in generalized derivation identities, we work with generalized derivation triples and quasiderivation pairs. In this framework, these derivation-type maps are treated as mathematical operators encoding infinitesimal symmetries of the underlying graded algebraic structure. The triple space is a Lie superalgebra, and quasiderivation pairs fit into a short exact sequence of Lie superalgebras whose kernel measures precisely the ambiguity of the auxiliary map. We prove GDerα(L)=QDerα(L)+QCentα(L), and establish the corresponding Ω-compatible decomposition for balanced generalized derivations. We then construct a three-layer envelope for generalized derivation triples and a two-layer envelope for quasiderivation pairs, obtaining injective Lie-superalgebra embeddings into α-commuting derivation algebras. A structural obstruction shows that, when α2≠0, the original nonzero Ω cannot simply be copied to the active layers of these two-step envelopes as their structural form. For centerless algebras, the full derivation superalgebra of the two-layer envelope is determined as an explicit semidirect product. Finally, a structure-constant criterion reduces the pair and triple spaces to homogeneous linear systems. For the regular family Hτ, τ∈C×∖{1}, we obtain dimDerατ=1 and dimQDerατ=dimGDerατ=4, while the Ω-compatible first-component space has dimension 3.
Related
- Derivations and quasi-derivations of Block Lie superalgebras S(q)
- n-Derivations of Lie Superalgebras: Structure, Decomposition, and Central Extensions
- Non-existence of Negative Weight Derivations of the Moduli Algebras of Isolated Weighted Homogeneous Hypersurface Singularities
- A characterization of generalized derivations and biderivations on some simple algebras
- Commutativity Criteria and Structural Properties of Prime Near-Rings Admitting Right Generalized Derivations on Specific Ideals
BibTeX
@article{suhail2026generalized,
title = {{Generalized Derivations and Canonical Envelopes of Complex α2-Weighted Ω-Hom-Lie Superalgebras}},
author = {Muntasir Suhail and Osman Abdalla Adam Osman and Mohammed Rabih},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203813},
url = {https://www.mdpi.com/2227-7390/14/20/3813},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Generalized Derivations and Canonical Envelopes of Complex α2-Weighted Ω-Hom-Lie Superalgebras
AU - Muntasir Suhail
AU - Osman Abdalla Adam Osman
AU - Mohammed Rabih
PY - 2026
DA - 2026/10/20
JO - Mathematics
DO - 10.3390/math14203813
UR - https://www.mdpi.com/2227-7390/14/20/3813
AB - We study generalized derivations of complex multiplicative α2-weighted Ω-Hom-Lie superalgebras. The chosen structural identity is shown to be stable under Yau twisting, and every regular algebra in this class untwists canonically to an Ω-Lie superalgebra. To retain the auxiliary operators that occur in generalized derivation identities, we work with generalized derivation triples and quasiderivation pairs. In this framework, these derivation-type maps are treated as mathematical operators encoding infinitesimal symmetries of the underlying graded algebraic structure. The triple space is a Lie superalgebra, and quasiderivation pairs fit into a short exact sequence of Lie superalgebras whose kernel measures precisely the ambiguity of the auxiliary map. We prove GDerα(L)=QDerα(L)+QCentα(L), and establish the corresponding Ω-compatible decomposition for balanced generalized derivations. We then construct a three-layer envelope for generalized derivation triples and a two-layer envelope for quasiderivation pairs, obtaining injective Lie-superalgebra embeddings into α-commuting derivation algebras. A structural obstruction shows that, when α2≠0, the original nonzero Ω cannot simply be copied to the active layers of these two-step envelopes as their structural form. For centerless algebras, the full derivation superalgebra of the two-layer envelope is determined as an explicit semidirect product. Finally, a structure-constant criterion reduces the pair and triple spaces to homogeneous linear systems. For the regular family Hτ, τ∈C×∖{1}, we obtain dimDerατ=1 and dimQDerατ=dimGDerατ=4, while the Ω-compatible first-component space has dimension 3.
SN - 2227-7390
ER -