# 2-generation of simple Lie algebras and free dense subgroups of algebraic groups

## TL;DR

This paper constructs generating pairs for simple Lie algebras over characteristic zero and uses these to produce infinite series of free, dense subgroups of certain simple algebraic groups, advancing understanding of their subgroup structures.

## Contribution

It introduces a method to generate simple Lie algebras with pairs and applies it to produce new free dense subgroups in classical algebraic groups.

## Key findings

- Constructed generating pairs for simple Lie algebras in characteristic zero.
- Produced infinite series of 2-generator Zariski dense subgroups.
- Demonstrated these subgroups are free of rank 2 in specific algebraic groups.

## Abstract

We construct generating pairs of simple Lie algebras in characteristic zero. We apply this construction to exhibit infinite series of 2-generator Zariski dense subgroups that are free of rank 2 of the simple algebraic groups SL(n, C), Sp(n, C), G_2(C).