The structure of the linearizer of a connected complex Lie group

TL;DR

This paper proves that the linearizer of a connected complex Lie group has a simplified structure, with the vector group component being trivial, extending Morimoto's theorem.

Contribution

It establishes that the linearizer of a connected complex Lie group lacks a vector group component, refining the structure described by Morimoto's theorem.

Findings

The linearizer of a connected complex Lie group has no vector group component.

The structure of the linearizer aligns with Morimoto's decomposition, excluding the vector part.

The result simplifies understanding the structure of complex Lie groups' linearizers.

Abstract

The Morimoto theorem states that each connected abelian complex Lie group $A$ can be decomposed into the direct product of a group on which all holomorphic functions are constant, finitely many copies of $\mathbb{C}^\times$ and a vector group. We prove that if $A$ is the complex linearizer of a connected complex Lie group then the last factor of the product is trivial.