Partial actions on reductive Lie algebras

In this paper, we study partial group actions on Lie algebras. We describe the structure of the inverse semigroup of all partial automorphisms (isomorphisms between ideals) of a finite-dimensional reductive Lie algebra. Also, we show that every partial group action on a finite-dimensional semisimple Lie algebra admits a globalization, unique up to isomorphism. As a consequence, we obtain that the globalization problem for partial group actions on reductive Lie algebra is equivalent to the globalization problem on its center.