Maximal ideals and representations of twisted forms of algebras

TL;DR

This paper characterizes the maximal ideals of twisted forms of algebra of currents over a ring extension and classifies their finite-dimensional simple modules, extending understanding of algebra representations.

Contribution

It provides a bijection between maximal ideals of the base ring and twisted forms, and classifies simple modules over these twisted algebras.

Findings

Maximal ideals of twisted forms correspond to those of the base ring.

Complete classification of finite-dimensional simple modules for twisted forms of Lie algebras.

Establishes a foundational link between algebraic structures and their module representations.

Abstract

Given a central simple algebra $\mathfrak{g}$ and a Galois extension of base rings $S/R$, we show that the maximal ideals of twisted $S/R$-forms of the algebra of currents $\mathfrak{g}(R)$ are in natural bijection with the maximal ideals of $R$. When $\mathfrak{g}$ is a Lie algebra, we use this to give a complete classification of the finite-dimensional simple modules over twisted forms of $\mathfrak{g}(R)$.