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Actions of Solvable Algebraic Groups on Central Simple Algebras

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Abstract

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.

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References

  1. Farb, B., Dennis, R.K.: Noncommutative Algebra. Springer, New York (1993)

    MATH  Google Scholar 

  2. Humphreys, J.: Linear Algebraic Groups, corrected second printing. Springer, New York (1981)

    Google Scholar 

  3. McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Wiley, Chichester (1987)

    MATH  Google Scholar 

  4. Reichstein, Z., Vonessen, N.: Torus actions on rings. J. Algebra 170, 781–804 (1994)

    Article  MATH  Google Scholar 

  5. Reichstein, Z., Vonessen, N.: Rational central simple algebras. Israel J. Math. 95, 253–280 (1996)

    MATH  Google Scholar 

  6. Reichstein, Z., Vonessen, N.: Group actions on central simple algebras: a geometric approach. J. Algebra 304, 1160–1192 (2006)

    Article  MATH  Google Scholar 

  7. Rowen, L.H.: Polynomial Identities in Ring Theory. Academic, New York (1980)

    MATH  Google Scholar 

  8. Springer, T.A.: Linear Algebraic Groups, 2nd edn. Birkhäuser, Boston (1998)

    MATH  Google Scholar 

  9. Vonessen, N.: Actions of linearly reductive groups on affine PI-algebras. Mem. Amer. Math. Soc. 414, (1989)

  10. Vonessen, N.: Actions of linearly reductive groups on PI-algebras. Trans. Amer. Math. Soc. 335, 425–442 (1993)

    Article  MATH  Google Scholar 

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  1. Department of Mathematical Sciences, University of Montana, Missoula, MT, 59812-0864, USA

    Nikolaus Vonessen

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  1. Nikolaus Vonessen
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Correspondence to Nikolaus Vonessen.

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Vonessen, N. Actions of Solvable Algebraic Groups on Central Simple Algebras. Algebr Represent Theor 10, 413–427 (2007). https://doi.org/10.1007/s10468-007-9052-7

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  • DOI: https://doi.org/10.1007/s10468-007-9052-7

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