Abstract:
Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these derivations act on a differential category. A differential Tannakian theory is developed. The main application is to the Galois theory of linear differential equations with parameters. Namely, we show the existence of a parameterized Picard-Vessiot extension and, therefore, the Galois correspondence for many differential fields with, possibly, non-differentially closed fields of constants, that is, fields of functions of parameters. Other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic, which will appear in a separate paper. This application is based on differential categories developed in the present paper, and not just differential algebraic groups and their representations.
H. Gillet was supported by the grants NSF DMS-0500762 and DMS-0901373. S. Gorchinskiy was supported by the grants RFBR 11-01-00145-a, NSh-4713.2010.1, MK-4881.2011.1, and AG Laboratory GU-HSE, RF government grant, ag. 11 11.G34.31.0023. A. Ovchinnikov was supported by the grants: NSF CCF-0952591 and PSC-CUNY No. 60001-40 41.
Lucia Di Vizio, Charlotte Hardouin, “Galois theories of q-difference equations: comparison theorems”, Confluentes Mathematici, 12:2 (2021), 11
Andrei Minchenko, Alexey Ovchinnikov, “Triviality of differential Galois cohomology of linear differential algebraic groups”, Communications in Algebra, 47:12 (2019), 5094
Lucia Di Vizio, “Action of an endomorphism on (the solutions of) a linear differential equation”, Publications mathématiques de Besançon. Algèbre et théorie des nombres, 2019, no. 1, 21
Andrei Minchenko, Alexey Ovchinnikov, “Calculating Galois groups of third-order linear differential equations with parameters”, Commun. Contemp. Math., 20:04 (2018), 1750038
Quentin Brouette, Françoise Point, “On differential Galois groups of strongly normal extensions”, Mathematical Logic Qtrly, 64:3 (2018), 155
Quentin Brouette, Greg Cousins, Anand Pillay, Francoise Point, “Embedded Picard–Vessiot extensions”, Communications in Algebra, 46:11 (2018), 4609
OMAR LEÓN SÁNCHEZ, ANAND PILLAY, “SOME DEFINABLE GALOIS THEORY AND EXAMPLES”, Bull. symb. log, 23:2 (2017), 145
Charlotte Hardouin, Andrei Minchenko, Alexey Ovchinnikov, “Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence”, Math. Ann., 368:1-2 (2017), 587
Anand Pillay, “The Picard–Vessiot theory, constrained cohomology, and linear differential algebraic groups”, Journal de Mathématiques Pures et Appliquées, 108:6 (2017), 809
Alexey Ovchinnikov, Michael Wibmer, “Tannakian Categories With Semigroup
Actions”, Can. j. math., 69:3 (2017), 687
Omar León Sánchez, Joel Nagloo, “On parameterized differential Galois extensions”, Journal of Pure and Applied Algebra, 220:7 (2016), 2549
Annette Bachmayr, David Harbater, Julia Hartmann, “Differential Galois groups over Laurent series fields”, Proc. London Math. Soc., 112:3 (2016), 455
Moshe Kamensky, Anand Pillay, “Interpretations and Differential Galois Extensions”, Int Math Res Notices, 2016:24 (2016), 7390
Carlos E. Arreche, “On the computation of the parameterized differential Galois group for a second-order linear differential equation with differential parameters”, Journal of Symbolic Computation, 75 (2016), 25
Annette Maier, “On the parameterized differential inverse Galois problem over k((t))(x)”, Journal of Algebra, 428 (2015), 43
Andrey Minchenko, Alexey Ovchinnikov, Michael F. Singer, “Reductive Linear Differential Algebraic Groups and the Galois Groups of Parameterized Linear Differential Equations”, International Mathematics Research Notices, 2015:7 (2015), 1733
Thomas Dreyfus, “A density theorem in parametrized differential Galois theory”, Pacific J. Math., 271:1 (2014), 87
Contact us: