Underlying varieties and group structures

TL;DR

This paper investigates how the underlying geometric or topological structure of algebraic and Lie groups influences or determines their algebraic or group-theoretic properties.

Contribution

It provides insights into the relationship between the underlying variety or manifold and the group structure, clarifying when the structure is uniquely determined.

Findings

The underlying variety can determine the algebraic group in certain cases.

The underlying manifold influences the group structure in real Lie groups.

Conditions under which the variety or manifold uniquely determines the group are identified.

Abstract

We explore to what extent the underlying variety of a connected algebraic group or the underlying manifold of a real Lie group determines its group structure.