Local derivations of the modular Lie algebra \(\mathfrak {sl}(2)\) to its all simple modules

Over a field of characteristic \(p>2\) $p>2$ , we determine all local derivations of the 3-dimensional simple Lie algebra \(\mathfrak {sl}(2)\) $\mathfrak{sl}(2)$ to its any finite-dimensional simple module \(L_{\chi }(\lambda )\) $L_{\chi }(\lambda )$ by use of the theory of systems of linear equations. The results show that only in the case of \(\chi =0, \lambda \in \{0, 2, \ldots , p-3, p-1\}\) $\chi =0,\lambda \in \{0,2,\dots ,p-3,p-1\}$ or \(\chi =0, \lambda =1, p\ne 3\) $\chi =0,\lambda =1,p\ne 3$ , every local derivation of \(\mathfrak {sl}(2)\) $\mathfrak{sl}(2)$ to \(L_{\chi }(\lambda )\) $L_{\chi }(\lambda )$ is a derivation. In particular, we determine the codimension of the derivation space in the local derivation space for \(\mathfrak {sl}(2)\) $\mathfrak{sl}(2)$ and any \(L_{\chi }(\lambda )\) $L_{\chi }(\lambda )$ .