Global Lie–Tresse theorem

Kruglikov, Boris; Lychagin, Valentin

doi:10.1007/s00029-015-0220-z

Cited by 83 publications

(125 citation statements)

References 40 publications

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“…Writing the differential invariants explicitly in terms of submanifold jet coordinates (x, u (n) ), these constraints give invariant differential equations that u = u(x) must satisfy. The proposition then follows from the fact that the differential module of differential syzygies restricted to the solution space of an invariant differential equation is finitely generated [22,Theorem 24]. REMARK 32.…”

Section: Group Foliationmentioning

confidence: 98%

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Group Foliation of Differential Equations Using Moving Frames

Thompson

Valiquette

2015

Forum math. Sigma

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We incorporate the new theory of equivariant moving frames for Lie pseudogroups into Vessiot's method of group foliation of differential equations. The automorphic system is replaced by a set of reconstruction equations on the pseudogroup jets. The result is a completely algorithmic and symbolic procedure for finding both invariant and noninvariant solutions of differential equations admitting a symmetry group.2010 Mathematics Subject Classification: 35B06, 53A55 (primary); 35C05, 58H05 (secondary)

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Section: Group Foliationmentioning

confidence: 98%

“…Working on the open subset of jet space where K = 0, one can replace the invariant total derivative operators D x and D y by the invariant Tresse derivatives D H and D J [22]. By the chain rule,…”

Section: (34)mentioning

confidence: 99%

Group Foliation of Differential Equations Using Moving Frames

Thompson

Valiquette

2015

Forum math. Sigma

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“…Are there sufficiently many such observables to separate points + on the physical phase spaceP of GR? The general mathematical context in which the answers must be sought is known as differential invariant theory [26,30,25]. Classical invariant theory is concerned with identifying functions on a G -space (a space with an action of a group G ) that are invariant under the G -action, these are the usual invariants.…”

Section: Gauge Invariance and Local Observables In Gravitational Theomentioning

confidence: 99%

Local and gauge invariant observables in gravity

Khavkine

2015

Class. Quantum Grav.

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Abstract. It is well known that general relativity (GR) does not possess any non-trivial local (in a precise standard sense) and diffeomorphism invariant observables. We propose a generalized notion of local observables, which retain the most important properties that follow from the standard definition of locality, yet is flexible enough to admit a large class of diffeomorphism invariant observables in GR. The generalization comes at a small price, that the domain of definition of a generalized local observable may not cover the entire phase space of GR and two such observables may have distinct domains. However, the subset of metrics on which generalized local observables can be defined is in a sense generic (its open interior is non-empty in the Whitney strong topology). Moreover, generalized local gauge invariant observables are sufficient to separate diffeomorphism orbits on this admissible subset of the phase space. Connecting the construction with the notion of differential invariants, gives a general scheme for defining generalized local gauge invariant observables in arbitrary gauge theories, which happens to agree with well-known results for Maxwell and Yang-Mills theories.

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“…It is well known that the algebra of differential invariants of a Lie transformation group, [17,39], or (modulo technical hypotheses) a Lie pseudo-group, [31,46], is generated from a finite number of low order generating differential invariants through successive application of the operators of invariant differentiation. The construction of the generating differential invariants, the invariant differential operators, and the identities (syzygies and recurrence relations among them) can be completely systematized through the symbolic calculus provided by the equivariant method of moving frames, [17,29,33,45].…”

Section: Invariant Differential Forms and Differential Operatorsmentioning

confidence: 99%

Invariants of objects and their images under surjective maps

Kogan

Olver

2015

Lobachevskii J Math

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We examine the relationships between the differential invariants of objects and of their images under a surjective map. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the image, and the case when only a proper subgroup of the entire group acts projectably. In the former case, we establish a constructible isomorphism between the algebra of differential invariants of the images and the algebra of fiber-wise constant (gauge) differential invariants of the objects. In the latter case, we describe residual effects of the full transformation group on the image invariants. Our motivation comes from the problem of reconstruction of an object from multiple-view images, with central and parallel projections of curves from three-dimensional space to the two-dimensional plane serving as our main examples.

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Global Lie–Tresse theorem

Abstract: We prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.

Cited by 83 publications

References 40 publications

Group Foliation of Differential Equations Using Moving Frames

Group Foliation of Differential Equations Using Moving Frames

Local and gauge invariant observables in gravity

Invariants of objects and their images under surjective maps

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