Twisted product of Lie groups

TL;DR

This paper introduces the concept of twisted products of groups as a generalization of semidirect products, explores conditions for their group structure, and examines their Lie algebra and scalar curvature properties.

Contribution

It defines twisted products of groups, establishes conditions for their group structure, and analyzes the Lie algebra and scalar curvature in the context of Lie groups.

Findings

Twisted product of a group with itself is a group iff the initial group is metabelian.

Constructs Lie algebra for twisted product of Lie groups.

Scalar curvature of the twisted product depends on the initial group's scalar curvature.

Abstract

In this article we define the twisted product of groups as the generalization of the semidirect product of groups. We will find the necessary and sufficient condition in order that the twisted product of groups to be a group. In particular, for two copies of the same group, the twisted product of group by itself through the action of inner automorphisms is a group if and only if the initial group is a metabelian group. Further we will construct Lie algebra for Lie group of a twisted product of Lie groups. In the case of twisted product of Lie group by itself by means of the action of inner automorphisms we find the dependence of the scalar curvature for resulting Lie group on the scalar curvature for initial Lie group.