Sum-product for real Lie groups

TL;DR

This paper establishes a sum-product theorem for certain Lie group representations and uses it to derive a product theorem for perfect Lie groups, advancing understanding of algebraic expansion properties.

Contribution

It introduces a discretized sum-product theorem for Lie group representations without trivial components, leading to a new product theorem in perfect Lie groups.

Findings

Proved a sum-product theorem for specific Lie group representations.

Derived a product theorem for perfect Lie groups.

Enhanced understanding of algebraic expansion in Lie groups.

Abstract

We prove a discretized sum-product theorem for representations of Lie groups whose Jordan-H\"older decomposition does not contain the trivial representation. This expansion result is used to derive a product theorem in perfect Lie groups.