Actions of solvable algebraic groups on central simple algebras

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Let k be an algebraically closed base field of arbitrary characteristic. In this paper, we study actions of a connected solvable linear algebraic group G on a central simple algebra Q. The main result is the following: Q can be split G-equivariantly by a finite-dimensional splitting field, provided that G acts "algebraically," i.e., provided that Q contains a G-stable order on which the action is rational. As an application, it is shown that rational torus actions on prime PI-algebras are induced by actions on commutative domains.

Original languageEnglish
Pages (from-to)413-427
Number of pages15
JournalAlgebras and Representation Theory
Issue number5
StatePublished - Oct 2007


  • Algebraic action
  • Central simple algebra
  • Division algebra
  • Group action
  • PI-algebra
  • Rational action
  • Solvable linear algebraic group
  • Splitting field


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