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Differential Galois Groups of Differential Central Simple Algebras and their Projective Representations

Manujith K. Michel, Varadharaj R. Srinivasan

Abstract

Let $F$ be a $δ-$field (differential field) of characteristic zero with an algebraically closed field of constants $F^δ$, $A$ be a $δ-F-$central simple algebra, $K$ be a Picard-Vessiot extension for the $δ-F-$module $A$ and $\mathscr G(K|F)$ be the $δ-$Galois group of $K$ over $F.$ We prove that a $δ-$field extension $L$ of $F,$ having $F^δ$ as its field of constants, splits the $δ-F-$central simple algebra $A$ if and only if the $δ-$field $K$ embeds in $L.$ We then extend the theory of $δ-F-$matrix algebras over a $δ-$field $F,$ put forward by Magid & Juan (2008), to arbitrary $δ-F-$central simple algebras. In particular, we establish a natural bijective correspondence between the isomorphism classes of $δ-F-$central simple algebras of dimension $n^2$ over $F$ that are split by the $δ-$field $K$ and the classes of inequivalent representations of the algebraic group $\mathscr G(K|F)$ in $\mathrm{PGL}_n(F^δ).$ We show that $\mathscr G(K|F)$ is a reductive or a solvable algebraic group if and only if $A$ has certain kinds of $δ-$right ideals.

Differential Galois Groups of Differential Central Simple Algebras and their Projective Representations

Abstract

Let be a field (differential field) of characteristic zero with an algebraically closed field of constants , be a central simple algebra, be a Picard-Vessiot extension for the module and be the Galois group of over We prove that a field extension of having as its field of constants, splits the central simple algebra if and only if the field embeds in We then extend the theory of matrix algebras over a field put forward by Magid & Juan (2008), to arbitrary central simple algebras. In particular, we establish a natural bijective correspondence between the isomorphism classes of central simple algebras of dimension over that are split by the field and the classes of inequivalent representations of the algebraic group in We show that is a reductive or a solvable algebraic group if and only if has certain kinds of right ideals.
Paper Structure (14 sections, 15 theorems, 82 equations)

This paper contains 14 sections, 15 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Picard-Vessiot Theory for Differential Modules.
  4. Construction of a Picard-Vessiot Extension
  5. The Tannakian Formalism in Differential Galois Theory
  6. Completely reducible modules.
  7. minimal differential splitting fields of differential central simple algebras
  8. Trivializations and splittings
  9. The $\delta-F-$Brauer Monoid.
  10. Projective Representations of Differential Galois Groups
  11. Complete reduciblity and solvability
  12. Derivations stabilizing maximal subfields
  13. Liouvillain field extensions
  14. Groups with trivial torsor.

Key Result

Theorem 1.1

Let $F$ be a $\delta-$field, $F^\delta$ be algebraically closed and $A$ be a $\delta-F-$central simple algebra. Let $L$ be a $\delta-$field extension of $F$ with $F^{\delta}=L^{\delta}$ and $K$ be a Picard-Vessiot extension for the $\delta-F-$module $A.$

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.1.1
  • proof
  • ...and 22 more