On the universal <inline-formula><tex-math id="M1">$ \alpha $</tex-math></inline-formula>-central extensions of the semi-direct product of Hom-preLie algebras

Abstract

&lt;abstract&gt;&lt;p&gt;We study Hom-actions, semidirect product and describe the relation between semi-direct product extensions and split extensions of Hom-preLie algebras. We obtain the functorial properties of the universal &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \alpha $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-central extensions of &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \alpha $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-perfect Hom-preLie algebras. We give that a derivation or an automorphism can be lifted in an &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \alpha $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-cover with certain constraints. We provide some necessary and sufficient conditions about the universal &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \alpha $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-central extension of the semi-direct product of two &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ \alpha $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-perfect Hom-preLie algebras.&lt;/p&gt;&lt;/abstract&gt;