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Isomonodromic differential equations and differential categories. (English) Zbl 1332.12011

Summary: We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under a filtered-linearly closed assumption on the field of functions of parameters. Our result implies that one does not need to solve any nonlinear differential equations to test isomonodromicity anymore. This result cannot be further strengthened by weakening the requirement on the parameters as we show by giving a counterexample. Also, we show that isomonodromicity is equivalent to conjugacy to constants of the associated parameterized differential Galois group, extending a result of P. Cassidy and M. Singer, which we also prove categorically. We illustrate our main results by a series of examples, using, in particular, a relation between the Gauss-Manin connection and parameterized differential Galois groups.

MSC:

12H20 Abstract differential equations
13N10 Commutative rings of differential operators and their modules
20G05 Representation theory for linear algebraic groups
34M56 Isomonodromic deformations for ordinary differential equations in the complex domain
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
58A12 de Rham theory in global analysis
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