Rectangular representations and $\lambda$-independence of algebraic monodromy groups

TL;DR

This paper introduces rectangular representations of complex semisimple Lie algebras, classifies faithful examples, and applies these results to establish new $mbda$-independence properties of algebraic monodromy groups in Galois representations.

Contribution

It defines rectangular representations, classifies faithful ones, and applies this to prove $mbda$-independence of algebraic monodromy groups in compatible Galois systems.

Findings

Classified all faithful rectangular representations of complex semisimple Lie algebras.

Established new $mbda$-independence results for algebraic monodromy groups.

Connected representation theory with Galois representation monodromy properties.

Abstract

Let $\mathfrak g$ be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of $\mathfrak g$ to be rectangular and completely classify faithful rectangular representations. As an application, we obtain new $\lambda$-independence results on the algebraic monodromy groups of compatible systems of $\lambda$-adic Galois representations of number fields.