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31
Differential Invariant Algebras of Lie Pseudo–Groups
, 2012
"... The aim of this paper is to describe, in as much detail as possible and constructively, the structure of the algebra of differential invariants of a Lie pseudogroup acting on the submanifolds of an analytic manifold. Under the assumption of local freeness ofasuitablyhighorder prolongationofthepse ..."
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The aim of this paper is to describe, in as much detail as possible and constructively, the structure of the algebra of differential invariants of a Lie pseudogroup acting on the submanifolds of an analytic manifold. Under the assumption of local freeness ofasuitablyhighorder prolongationofthepseudogroup action, wedevelop computational algorithms for locating a finite generating set of differential invariants, a complete system of recurrence relations for the differentiated invariants, and a finite system of generating differential syzygies among the generating differential invariants. In particular, if the pseudogroup acts transitively on the base manifold, then the algebra of differential invariants is shown to form a rational differential algebra with noncommuting derivations. The essential features of the differential invariant algebra are prescribed by a pair of commutative algebraic modules: the usual symbol module associated with the infinitesimal determining system of the pseudogroup, and a new “prolonged symbol module” constructed from the symbols of the annihilators of the prolonged pseudogroup generators. Modulo low order complications, thegenerating differential invariants and differential syzygies are in onetoone correspondence with the algebraic generators and syzygies of an invariantized version of the prolonged symbol module. Our algorithms and proofs are all constructive, and rely oncombining the movingframe approach developed inearlier papers with Gröbner basis algorithms from commutative algebra.
MaurerCartan Forms and the Structure of Lie Pseudo–Groups
, 2005
"... This paper begins a series devoted to developing a general and practical theory of moving frames for infinitedimensional Lie pseudogroups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations fo ..."
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Cited by 25 (10 self)
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This paper begins a series devoted to developing a general and practical theory of moving frames for infinitedimensional Lie pseudogroups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for a pseudogroup. Our approach is completely explicit and avoids reliance on the theory of exterior differential systems and prolongation. The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the pseudogroup on submanifolds. The third paper [61] will apply Gröbner basis methods to prove a fundamental theorem on the freeness of pseudogroup actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera for generating systems of differential invariants and also their syzygies. Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles, and solving equivalence and symmetry problems arising in geometry and physics.
Algorithms for differential invariants of symmetry groups of differential equations
 Found. Comput. Math
"... Abstract. We present new computational algorithms, based on equivariant moving frames, for classifying the differential invariants of Lie symmetry pseudogroups of differential equations and establishing the structure of the induced differential invariant algebra. The Korteweg–deVries and Kadomtsev– ..."
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Abstract. We present new computational algorithms, based on equivariant moving frames, for classifying the differential invariants of Lie symmetry pseudogroups of differential equations and establishing the structure of the induced differential invariant algebra. The Korteweg–deVries and Kadomtsev–Petviashvili equations are studied to illustrate these methods. 1.
Recursive Moving Frames
, 2011
"... A recursive algorithm for the equivariant method of moving frames, for both finitedimensional Lie group actions and Lie pseudogroups, is developed and illustrated by several examples of interest. The recursive method enables one avoid unwieldy symbolic expressions that complicate the treatment of ..."
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Cited by 5 (2 self)
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A recursive algorithm for the equivariant method of moving frames, for both finitedimensional Lie group actions and Lie pseudogroups, is developed and illustrated by several examples of interest. The recursive method enables one avoid unwieldy symbolic expressions that complicate the treatment of large scale applications of the equivariant moving frame method.
Algebraic and Differential Invariants
 FOUNDATIONS OF COMPUTATIONAL MATHEMATICS, BUDAPEST 2011, LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES (403)
, 2011
"... This article highlights a coherent series of algorithmic tools to compute and work with algebraic and differential invariants. ..."
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Cited by 5 (5 self)
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This article highlights a coherent series of algorithmic tools to compute and work with algebraic and differential invariants.
Lie completion of pseudogroups
, 2009
"... By far the most important class of pseudogroups, for both theory and in essentially all applications, are the Lie pseudogroups. In this paper, we propose a definition of the Lie completion of a regular pseudogroup, and establish some of its basic properties. In particular, a pseudogroup and its ..."
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By far the most important class of pseudogroups, for both theory and in essentially all applications, are the Lie pseudogroups. In this paper, we propose a definition of the Lie completion of a regular pseudogroup, and establish some of its basic properties. In particular, a pseudogroup and its Lie completion have exactly the same differential invariants and invariant differential forms. Thus, for practical purposes, one can exclusively work within
Solving local equivalence problems with the equivariant moving frame method
"... Abstract. Given a Lie pseudogroup action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate mod ..."
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Abstract. Given a Lie pseudogroup action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate modifications and assumptions, the equivariant moving frame constructions extend to submanifold jets where the pseudogroup does not act freely at any order. Once this is done, we review the solution to the local equivalence problem of submanifolds within the equivariant moving frame framework. This offers an alternative approach to Cartan’s equivalence method based on the theory of Gstructures. Key words: differential invariant; equivalence problem; Maurer–Cartan form; moving frame
Differential Invariant Algebras
"... Abstract. The equivariant method of moving frames provides an algorithmic procedure for determining and analyzing the structure of algebras of differential invariants for both finitedimensional Lie groups and infinitedimensional Lie pseudogroups. This paper surveys recent developments, including ..."
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Abstract. The equivariant method of moving frames provides an algorithmic procedure for determining and analyzing the structure of algebras of differential invariants for both finitedimensional Lie groups and infinitedimensional Lie pseudogroups. This paper surveys recent developments, including a few surprises and several open questions. 1. Introduction. Differential invariants are the fundamental building blocks for constructing invariant differential equations and invariant variational problems, as well as determining their explicit solutions and conservation laws. The equivalence, symmetry and rigidity properties of submanifolds are all governed by their differential invariants. Additional applications
Invariants of Solvable Lie Algebras with Triangular Nilradicals and Diagonal Nilindependent Elements
, 706
"... The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on ..."
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The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on Cartan’s method of moving frames. 1