Partial actions on reductive Lie algebras

Abstract

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.

Keywords:

• Globalization
• Lie algebras
• partial actions

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 17B20
• Secondary: 17B99
• 20M18

Acknowledgments

The author is very grateful to Misha Dokuchaev for the time spent in his helpful conversations and for his many suggestions.

Additional information

Funding

Rodríguez was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo, processo: 2019/08659-8.