Mathematics

We show that a differential algebraic group can be filtered by a finite subnormal series of differential algebraic groups such that successive quotients are almost simple, that is have no normal subgroups of the same type. We give a uniqueness result, prove several properties of almost simple groups and, in the ordinary differential case, classify almost simple linear differential algebraic groups.

Assumptions and notation. 1. The differential Zariski topology on affine space. 892 2. Characteristic sets of prime differential ideals. 894 3. The Zariski topology on affine space. 896 4. Extending the universal differential field. 897 5. Differential rational functions. 898 6 ...

Assumptions and notation. 1. The differential Zariski topology on affine space. 892 2. Characteristic sets of prime differential ideals. 894 3. The Zariski topology on affine space. 896 4. Extending the universal differential field. 897 5. Differential rational functions. 898 6 ...

We show that a differential algebraic group can be filtered by a finite subnormal series of differential algebraic groups such that successive quotients are almost simple, that is have no normal subgroups of the same type. We give a uniqueness result, prove several properties of almost simple groups and, in the ordinary differential case, classify almost simple linear differential algebraic

The differential algebraic group structures on the affine line and plane are classified. The additive group G a {G_a} of the coefficient field is the only differential algebraic group structure on the line. Every differential algebraic group with underlying set in the plane is unipotent and is isomorphic to a group whose law of composition is defined by the formula \[ ( u 1 , u 2 ) ( v 1 , v 2 ) = ( u 1 + v 1 , u 2 + v 2 + f ( u 1 , v 1 ) ) , ({u_1},{u_2})({v_1},{v_2}) = ({u_1} + {v_1},{u_2} + {v_2} + f({u_1},{v_1})), \] where f is a 2-cocycle of G a {G_a} into G a {G_a} .

A class of infinite-dimensional Lie algebras over the field K \mathcal {K} of constants of a universal differential field U \mathcal {U} is studied. The simplest case, defined by homogeneous linear differential equations, is analyzed in detail, and those with underlying set U × U \mathcal {U}\, \times \,\mathcal {U} are classified.

We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We present the basic constructions and results, give examples, discuss how isomonodromic families fit into this theory and show how results from the theory of linear differential algebraic groups may be used to classify systems of second order linear differential equations.

Selected Titles in This Series Volume 12 Ellis Kolchin: Selected Works of Ellis Kolchin with Commentary (Hyman Bass. Alexandra Buium, and Phyllis J. Cassidy, Editors), 1999 11 VS Varadarajan: The Selected Works of VS Varadarajan, 1999 10 Maurice Auslander: Selected Works of ...