Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations

Abstract

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending existing results, which were obtained for SL(2) in the case of just one derivation. As an application of the above bound, we develop an algorithm that tests whether the parameterized differential Galois group of a system of linear differential equations is reductive and, if it is, calculates it.

61 pages

References in corpus (10)

• Parameterized Picard-Vessiot extensions and Atiyah extensions
• Tannakian categories, linear differential algebraic groups, and parameterized linear differential equations
• Isomonodromic differential equations and differential categories
• Unipotent differential algebraic groups as parameterized differential Galois groups
• Tannakian approach to linear differential algebraic groups
• A density theorem for parameterized differential Galois theory
• A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function
• Existence of $\partial$-parameterized Picard-Vessiot extensions over fields with algebraically closed constants
• Computing the differential Galois group of a one-parameter family of second order linear differential equations
• Computing the Galois group of some parameterized linear differential equation of order two

Cited by in corpus (14)

• Isomonodromic differential equations and differential categories
• A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function
• Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence
• Computing the differential Galois group of a parameterized second-order linear differential equation
• Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation
• Difference algebraic relations among solutions of linear differential equations
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
• Computing differential Galois groups of second-order linear $q$-difference equations
• Difference integrability conditions for parameterized linear difference and differential equations
• Finiteness properties of affine difference algebraic groups
• Parameterized Differential Equations over k((t))(x)
• Parallel Telescoping and Parameterized Picard--Vessiot Theory
• Calculating Galois groups of third order linear differential equations with parameters
• Some applications of the parameterized Picard-Vessiot theory