On the Invariants of Matrices and the Embedding Problem

TL;DR

This paper investigates the properties of matrix invariants over infinite fields, demonstrating that certain algebraic structures with polynomial norms relate to matrix invariants through the Cayley-Hamilton theorem.

Contribution

It establishes a connection between algebras with polynomial norms satisfying Cayley-Hamilton-like conditions and quotients of invariant rings of matrices.

Findings

R is a quotient of the invariant ring of matrices under conjugation

Polynomial norms satisfying Cayley-Hamilton analogue imply algebraic structure constraints

Provides a new perspective on the embedding problem for algebras

Abstract

Let K be an infinite field and let R be a K-algebra endowed with a homogeneous polynomial norm N of degree n. If N satisfies a formal analogue of the Cayley-Hamilton Theorem the we will show that R is a quotient of the ring of the invariants of several square matrices of order n under the simultaneous conjugation.