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Computing the Lie algebra of the differential Galois group: the reducible case

Thomas Dreyfus, Jacques-Arthur Weil

Abstract

In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a Kolchin-Kovacic reduced form. We combine this with other reduction results to propose a general algorithm for computing a reduced form of a general linear differential system. In particular, this provides directly the Lie algebra of the differential Galois group without an a priori computation of this Galois group.

Computing the Lie algebra of the differential Galois group: the reducible case

Abstract

In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a Kolchin-Kovacic reduced form. We combine this with other reduction results to propose a general algorithm for computing a reduced form of a general linear differential system. In particular, this provides directly the Lie algebra of the differential Galois group without an a priori computation of this Galois group.

Paper Structure

This paper contains 30 sections, 18 theorems, 116 equations, 2 figures.

Table of Contents

  1. Differential Galois Theory and Reduced Forms
  2. The Base Field
  3. Differential Galois Theory
  4. Reduced Forms of Linear Differential Systems
  5. Decomposition and Flags for the Off-Diagonal Part
  6. The Off-Diagonal Algebra $\mathfrak{gl}_{\mathrm{sub}}$
  7. The Shape of the Reduction Matrix
  8. The Adjoint Action $\Psi = [A_{\mathrm{diag}} (x),\bullet]$
  9. Decomposition of $\mathfrak{gl}_{\mathrm{sub}}$ into $\Psi$-spaces
  10. Isotypical decomposition of $\mathfrak{gl}_{\mathrm{sub}}$ into $\Psi$-spaces
  11. The flag decomposition of an indecomposable $\Psi$-space
  12. Examples of Decomposition
  13. The "$SO_3\times SL_2$" Example.
  14. The "$SO_3\times B_2$" Example.
  15. The "$B_3\times B_2$" Example.
  16. ...and 15 more sections

Key Result

Lemma 1.4

Let $\mathfrak{n}\subset \mathcal{M}_n\left(\mathcal{C}\right)$ be a $\mathcal{C}$-vector space of lower triangular matrices with zero entries on the diagonal. Assume that, for all $N,N'\in \mathfrak{n}$, $N\cdot N'=(0)$. Then, ${U:=\{{\mathrm{Id}}_n+N, N\in \mathfrak{n}\}}$ is a connected algebrai

Figures (2)

  • Figure 1: The isotypical flag of the nilpotent example.
  • Figure 2: The action of the adjoint map on the isotypical flag of the nilpotent example $\S \ref{['nilpotent-example-continued']}$. The spaces $W_1$ (left), $W_2$ (center) and $W_3$ (right) satisfy $\mathfrak{gl}_{\mathrm{sub}}=W_1\oplus W_2 \oplus W_3$. The red rectangles correspond to the part that we get rid of via the reduction matrix, and the blue rectangles correspond to what will remain in the reduced matrix.

Theorems & Definitions (57)

  • Remark 1.1
  • Example 1.2
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Definition 1.6
  • Proposition 1.7: Kolchin-Kovacic reduction theorem
  • Definition 1.8
  • Remark 1.9
  • ...and 47 more