On some derivations of Lie conformal superalgebras

Abstract: Let $\mathcal{R}$ be a Lie conformal superalgebra. In this paper, we first investigate the conformal derivation algebra $CDer(\mathcal{R})$, the conformal triple derivation algebra $CTDer(\mathcal{R})$, and the generalized conformal triple derivation algebra $GCTDer(\mathcal{R})$. Moreover, we determine the connection of these derivation algebras. Next, we give a complete classification of the (generalized) conformal triple derivation algebra on all finite simple Lie conformal superalgebras. More specifically, $CTDer(\mathcal{R})=CDer(\mathcal{R})$, where $\mathcal{R}$ is a finite simple Lie conformal superalgebra, but for $GCTDer(\mathcal{R})$, we obtain a conclusion that is closely related to $CDer(\mathcal{R})$. Furthermore, we evaluate the $(\varPhi, \varPsi)$-Lie triple derivations on Lie conformal superalgebra, where $\varPhi$ and $\varPsi$ are associated automorphism of $\phi_{x}\in gc(\mathcal R)$. We evaluated some fundamental properties of $(\varPhi, \varPsi)$- Lie triple derivations. Later, we introduce the definition of $(A, B, C, D)$-derivation on Lie conformal superalgebra. We obtain the relationships between the generalized conformal triple derivations and the conformal $(A, B, C, D)$-derivations on Lie conformal superalgebra. Finally, we have presented the triple homomorphism of Lie conformal superalgebras.