#### Vol. 7, No. 2, 2013

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Maximal ideals and representations of twisted forms of algebras

### Michael Lau and Arturo Pianzola

Vol. 7 (2013), No. 2, 431–448
##### 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 $\mathfrak{g}\left(R\right)$.

##### Keywords
Galois descent, maximal ideals, finite-dimensional modules, multiloop algebras, twisted forms
##### Mathematical Subject Classification 2010
Primary: 17B10
Secondary: 17B67, 12G05, 17A60
##### Milestones
Received: 7 November 2011
Accepted: 3 March 2012
Published: 25 April 2013
##### Authors
 Michael Lau Département de mathématiques et de statistique Université Laval Pavillon Vachon (Local 1056) 1045 av. de la Médecine Québec, QC G1V0A6 Canada Arturo Pianzola Department of Mathematical and Statistical Sciences University of Alberta 632 Central Academic Building Edmonton, AB T6G2G1 Canada Centro de Altos Estudios en Ciencias Exactas Avenida de Mayo 866, (1084) Buenos Aires Argentina