Galois closures of non-commutative rings and an application to Hermitian representations

TL;DR

This paper extends the concept of Galois closures from commutative to non-commutative rings, demonstrating their properties and applying them to construct various representations in arithmetic invariant theory.

Contribution

It generalizes Galois closures to non-commutative rings and shows their compatibility with base change and product formulas, enabling new applications in representation theory.

Findings

Non-commutative Galois closures commute with base change

They satisfy a product formula

Application to constructing representations in arithmetic invariant theory

Abstract

Galois closures of commutative rank n ring extensions were introduced by Bhargava and the second author. In this paper, we generalize the construction to the case of non-commutative rings. We show that non-commutative Galois closures commute with base change and satisfy a product formula. As an application, we give a uniform construction of many of the representations arising in arithmetic invariant theory, including many Vinberg representations.