Differential Algebraic Groups of Finite Dimension
Differential algebraic groups were introduced by P. Cassidy and E. Kolchin and are, roughly speaking, groups defined by algebraic differential equations in the same way as algebraic groups are groups defined by algebraic equations. The aim of the book is
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Alexandru Buium
Differential Algebraic Groups of Finite Dimension
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Alexandru Buium Institute of Mathematics of the Romanian Academy P. O. Box 1-764 RO-70700 Bucharest, Romania
Mathematics Subject Classification (1991): l2H05, l4Ll7, l4KIO, 14G05, 20F28
ISBN 3-540-55181-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55181-6 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 - Printed on acid-free paper
INTRODucnON 1. Scope Differential algebraic groups are defined roughly speaking as "groups of solutions of algebraic differential equations" in the same way in which algebraic groups are defined as "groups of solutions of algebraic equations". They were introduced in modern literature by their pre-history goes back however to classical work of S. Lie, Cassidy [C i] and Kolchin [K Z]; E. Cartan and J.F. Ritt, Let's contemplate a few examples before giving the formal definition (for which we send to Section 3 of this Introduction). Start with any linear differential equation: (1)
where the unknown y and the coefficients a are, say, meromorphic functions of the complex i variable t, The difference of any two solutions of (1) is again a solution of (I); so the solutions of (I) form a group with respect to addition. This provides a first example of "differential algebraic group". SimiJarily, consider the system
(Z)
XY - I = O { yy" _ (y')Z + ayy' = 0
where the unknowns x,y and the coefficient a are once again meromorphic in t, This system (extracted from a paper of Cassidy [C ]) has the property that the quotient (x/x
y/YZ) of Z' 4 any two solutions (x I'YI)' (xZ,yZ) is again a solution; so the solutions of (Z) form a group with respect to the multiplicative group of the hyperbola xy - I
=0
and we are led to another
example of "differential algebraic group". Examples of a more subtle nature are provided by the systems
z
(3a)
y - xtx - l)(x - c) {
x"y
- x'y' + ax'y
=0
=0
z
y - x(x - l)(x - t) = 0
(3b)
{
3
Z
2
-y - 2(2t - l)(x - t) x'y + Zt(t - I)(x - t) (x"y - Zx'y') = 0
where the unknowns x, yare stiU merom orphic functions in t, the coefficient a is meromorphic in t and c
0,1 is a constant in C. These systems (of which (3b) is extracted f
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