Maximal ideals and representations of twisted forms of algebras

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 $\mathfrak{g}\left(R\right)$ 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 $\mathfrak{g}\left(R\right)$.