Smooth functions invariant under the action of a compact lie group

Schwarz, Gerald W.

doi:10.1016/0040-9383(75)90036-1

Cited by 391 publications

(304 citation statements)

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“…Schwarz [9] proves that in general for any compact Lie group Γ with an orthogonal action on R n , if the algebra of Γ-invariant polynomials is generated by ρ 1 , . .…”

Section: Linear Equivalence and Ode-equivalencementioning

confidence: 99%

Linear equivalence and ODE-equivalence for coupled cell networks

Dias

Stewart

2005

Nonlinearity

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Coupled cell systems are systems of ODEs, defined by 'admissible' vector fields, associated with a network whose nodes represent variables and whose edges specify couplings between nodes. It is known that non-isomorphic networks can correspond to the same space of admissible vector fields. Such networks are said to be 'ODE-equivalent'. We prove that two networks are ODE-equivalent if and only if they determine the same space of linear vector fields; moreover, the variable associated with each node may be assumed 1-dimensional for that purpose. We briefly discuss the combinatorics of the resulting linear algebra problem.

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“…Schwarz [9] proves that in general for any compact Lie group Γ with an orthogonal action on R n , if the algebra of Γ-invariant polynomials is generated by ρ 1 , . .…”

Section: Linear Equivalence and Ode-equivalencementioning

confidence: 99%

Linear equivalence and ODE-equivalence for coupled cell networks

Dias

Stewart

2005

Nonlinearity

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“…Since D(A, H, P) [D(A, H, P; C)] is the orbit space of a smooth finite-dimensional representation of a compact Lie group, it is a priori locally compact Hausdorff, and is thus homeomorphic to a semialgebraic subset of R d for some d [24]. The dimension of D(A, H, P) [D(A, H, P; C)] can then be defined as the dimension of this semialgebraic set.…”

Section: 3])mentioning

confidence: 99%

Moduli Spaces of Dirac Operators for Finite Spectral Triples

Ćaćić¹

2011

Quantum Groups and Noncommutative Spaces

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Abstract. The structure theory of finite real spectral triples developed by Krajewski and by Paschke and Sitarz is generalised to allow for arbitrary KOdimension and the failure of orientability and Poincaré duality, and moduli spaces of Dirac operators for such spectral triples are defined and studied. This theory is then applied to recent work by Chamseddine and Connes towards deriving the finite spectral triple of the noncommutative-geometric Standard Model.

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“…Define the set of smooth functions on M 0 as follows (see [9], or for the original source see [25]). …”

Section: Singular Reductionmentioning

confidence: 99%