Generalized Derivations and Canonical Envelopes of Complex α2-Weighted Ω-Hom-Lie Superalgebras

Source 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.