Expanding solitons to the Hermitian curvature flow on complex Lie groups

TL;DR

This paper studies the structure of complex Lie groups with left-invariant metrics that are expanding solitons under the Hermitian curvature flow, revealing their algebraic decomposition and how to construct such solitons.

Contribution

It provides a structural decomposition of complex Lie groups with expanding soliton metrics and a method to construct these solitons from their nilradicals.

Findings

Lie algebras decompose into semidirect products of reductive parts and nilradicals.

Restriction of the soliton metric to the nilradical is also an expanding algebraic soliton.

Method to construct expanding solitons from nilradicals' solitons.

Abstract

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on complex Lie groups starting from expanding solitons on their nilradicals.