In this paper, we present an algebraic description of the derivation classes and multiplicative Hom-structures of $ \mathfrak{so}(3) $. We also study $ \alpha^k $-derivations on multiplicative Hom-structures. Using the operator-matrix correspondence, explicit matrix representations for derivations, $ D $-derivations, and ($ F, D $)-derivations are derived by solving linear systems. We show that all nontrivial ($ F, D $)-derivations comprise scalar and skew-symmetric components. Three different multiplicative Hom-structures are explicitly parameterized, and, on this basis, we discuss $ \alpha^k $-derivations, obtaining the classification of $ \alpha^k $-derivations on the corresponding multiplicative Hom-Lie algebra $ \mathfrak{so}(3) $ in each case. Moreover, we show how each multiplicative Hom-structure gives rise to a Yau-twisted Hom-Lie bracket, and our classification of $ \alpha^k $-derivations corresponds to derivations of these twisted algebras. These results combine the algebraic and computational aspects of $ \mathfrak{so}(3) $, providing useful tools for applications in robotic kinematics and quantum symmetry breaking.
Citation: Yan Jiang, Ying Hou, Keli Zheng. Matrix representations of multiplicative Hom-structures and $ \alpha^k $-derivations on the real Lie algebra $ \mathfrak{so}(3) $[J]. AIMS Mathematics, 2026, 11(1): 2363-2383. doi: 10.3934/math.2026096
In this paper, we present an algebraic description of the derivation classes and multiplicative Hom-structures of $ \mathfrak{so}(3) $. We also study $ \alpha^k $-derivations on multiplicative Hom-structures. Using the operator-matrix correspondence, explicit matrix representations for derivations, $ D $-derivations, and ($ F, D $)-derivations are derived by solving linear systems. We show that all nontrivial ($ F, D $)-derivations comprise scalar and skew-symmetric components. Three different multiplicative Hom-structures are explicitly parameterized, and, on this basis, we discuss $ \alpha^k $-derivations, obtaining the classification of $ \alpha^k $-derivations on the corresponding multiplicative Hom-Lie algebra $ \mathfrak{so}(3) $ in each case. Moreover, we show how each multiplicative Hom-structure gives rise to a Yau-twisted Hom-Lie bracket, and our classification of $ \alpha^k $-derivations corresponds to derivations of these twisted algebras. These results combine the algebraic and computational aspects of $ \mathfrak{so}(3) $, providing useful tools for applications in robotic kinematics and quantum symmetry breaking.
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Yan Jiang, Ying Hou, Keli Zheng. Matrix representations of multiplicative Hom-structures and $ \alpha^k $-derivations on the real Lie algebra $ \mathfrak{so}(3) $[J]. AIMS Mathematics, 2026, 11(1): 2363-2383. doi: 10.3934/math.2026096
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