scholarly journals Typical differential dimension of the intersection of linear differential algebraic groups

1974 ◽  
Vol 32 (3) ◽  
pp. 476-487 ◽  
Author(s):  
William Yu Sit
Keyword(s):  
Algebraic Groups ◽  
Differential Dimension ◽  
Linear Differential ◽  
Linear Differential Algebraic Groups

scholarly journals Extensions of differential representations of SL2 and tori

2012 ◽  
Vol 12 (1) ◽  
pp. 199-224 ◽  
Author(s):  
Andrey Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Difference Equations ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Irreducible Representations ◽  
Direct Sums ◽  
Rational Representation ◽  
Linear Differential ◽  
Linear Differential Algebraic Groups

AbstractLinear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with ${\mathbf{SL} }_{2} $ and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of ${\mathbf{SL} }_{2} $. In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.

scholarly journals Finiteness Properties of Affine Difference Algebraic Groups

2020 ◽  
Author(s):  
Michael Wibmer
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Difference Equations ◽  
General Linear Group ◽  
Algebraic Groups ◽  
The Matrix ◽  
Algebraic Difference ◽  
The Difference ◽  
Linear Differential ◽  
Finiteness Properties

Abstract We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated.

scholarly journals DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

2015 ◽  
Vol 16 (1) ◽  
pp. 59-119 ◽  
Author(s):  
Lucia Di Vizio ◽  
Charlotte Hardouin ◽  
Michael Wibmer
Keyword(s):  
Differential Equations ◽  
Characteristic Zero ◽  
Galois Theory ◽  
Structure Theorem ◽  
Special Functions ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Linear Differential

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

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