Actions of solvable algebraic groups on central simple algebras

Nikolaus Vonessen

doi:10.1007/s10468-007-9052-7

Actions of solvable algebraic groups on central simple algebras

Nikolaus Vonessen

Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	413-427
Number of pages	15
Journal	Algebras and Representation Theory
Volume	10
Issue number	5
DOIs	https://doi.org/10.1007/s10468-007-9052-7
State	Published - Oct 2007

Funding

Acknowledgements The author gratefully acknowledges the support of the University of Montana and the hospitality of Zinovy Reichstein and of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

Funders
University of British Columbia

Keywords

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

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10.1007/s10468-007-9052-7

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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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N1 - Funding Information: Acknowledgements The author gratefully acknowledges the support of the University of Montana and the hospitality of Zinovy Reichstein and of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

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N2 - 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.

AB - 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.

KW - Algebraic action

KW - Central simple algebra

KW - Division algebra

KW - Group action

KW - PI-algebra

KW - Rational action

KW - Solvable linear algebraic group

KW - Splitting field

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VL - 10

SP - 413

EP - 427

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

IS - 5

ER -

Actions of solvable algebraic groups on central simple algebras

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