Doubling twisted pre-Lie algebras of finite topological spaces

Abstract

In this paper, we first study the twisted pre-Lie algebra of connected finite topological spaces. Then we construct the twisted pre-Lie structure on its doubling space. We describe the enveloping algebra of both twisted pre-Lie algebras, and we prove that the doubling space of connected topological spaces is a left module on the twisted pre-Lie algebra. We exhibit natural semi-products of both twisted pre-Lie algebras and both enveloping algebras respectively.