Generalized derivations and their embedding in $\omega$-hom-Lie algebras

In this study, we explored the algebraic structure of generalized derivations within finite-dimensional $\omega$-hom-Lie algebras over a field $K$, emphasizing their symmetry properties in nonassociative settings. We established a novel embedding theorem, proving that every compatible quasiderivation of an $\omega$-hom-Lie algebra can be represented as a compatible derivation in a larger, symmetrically constructed $\omega$-hom-Lie algebra. This result extended classical Lie algebra derivation theory, leveraging the skew-symmetric bilinear form $\omega$ and the homomorphism $\varphi$ to preserve structural symmetries. Additionally, we developed a computational algorithm, inspired by Gröbner basis techniques in commutative algebra for solving systems of polynomial equations arising from the derivation conditions, to explicitly calculate compatible generalized derivations and quasiderivations for all 3-dimensional non-Lie complex $\omega$-hom-Lie algebras with $\varphi =\mathrm{i}\mathrm{d}$ (i.e., the corresponding $\omega$-Lie algebras). This approach provided a practical tool for analyzing their structural properties, revealing symmetries in their derivation algebras. Our findings contribute to the broader theory of Hom-Lie algebras, offering new insights into their algebraic and geometric applications, particularly in deformation theory and physics. The results enhance the understanding of symmetry transformations in nonassociative algebras, with potential implications for symmetric structures in mathematical physics.