Representations of Compact Lie Groups

Bröcker, Theodor; Dieck, Tammo tom

doi:10.1007/978-3-662-12918-0

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“…We moreover assume that Γ is a compact group, i.e., after a coordinate transformation of R n it becomes a subset of O(n). Such continuous matrix groups are examples of Lie groups [2,26]. The easiest example of a Lie group is the group of rotations and reflections in the plane, O(2), or in three-dimensional space, O(3), and we will encounter these groups in the numerical continuation of periodic orbits of N -body problems, see Section 4.3 below.…”

Section: Numerical Continuation Of Relative Periodic Orbitsmentioning

confidence: 99%

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Numerical Continuation of Hamiltonian Relative Periodic Orbits

Wulff

Schebesch

2008

J Nonlinear Sci

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The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity.But as yet there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step towards a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our pathfollowing algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous Figure Eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating Eight family and compute the rotating choreographies bifurcating from it.

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Section: Numerical Continuation Of Relative Periodic Orbitsmentioning

confidence: 99%

“…To prove this notice that on one hand the Lie algebra of any Cartan subgroup is abelian (see [2]). On the other hand, let g (σ,µ) be abelian.…”

Section: Persistence Of Hamiltonian Rpos With Regular Drift-momentum mentioning

confidence: 99%

Numerical Continuation of Hamiltonian Relative Periodic Orbits

Wulff

Schebesch

2008

J Nonlinear Sci

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“…This result is of course well-known 4 in the sense that the representations of compact Lie groups are completely classified and that a proof "by inspection" is almost possible, see for example the end of [2]. However, we were not able to find a direct proof in textbooks on Lie groups.…”

Section: Natural Representationsmentioning

confidence: 71%

Wilson loops in the light of spin networks

LEVY¹

2004

Journal of Geometry and Physics

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If G is any finite product of orthogonal, unitary and symplectic matrix groups, then Wilson loops generate a dense subalgebra of continuous observables on the configuration space of lattice gauge theory with structure group G. If G is orthogonal, unitary or symplectic, then Wilson loops associated to the natural representation of G are enough.This extends a result of A. Sengupta [7]. In particular, our approach includes the case of even orthogonal groups. IntroductionOn a compact Lie group, the Peter-Weyl theorem asserts that the characters of irreducible representations generate a dense subalgebra of continuous functions invariant by adjunction. In lattice gauge theory, configuration spaces are powers of a Lie group on which another power of the same group acts, according to the geometry of a given graph and in a way which extends the adjoint action of the group on itself. Peter-Weyl theorem can be adapted to this situation and the functions that play the role of the characters are called spin networks. Despite the fact that spin networks were introduced about forty years ago in a physical context 1 , their importance in lattice gauge theory has been recognized rather recently [1]. In the mean time, another set of functions, easier to define, has been used as the standard set of observables: Wilson loops. However, it is not clear at all a priori that this set is complete, that is, that Wilson loops generate a dense subalgebra of continuous invariant functions on the configuration space. A. Sengupta has proved in [7] that it is true when the group is a product of odd orthogonal, unitary (and symplectic) groups. In this paper, an approach similar to that of Sengupta but with a little more classical invariant theory combined with the use of spin networks allows us to add even orthogonal groups to the list and, hopefully, to clarify the argument. * IRMA -7, rue René Descartes -F-67084 Strasbourg Cedex 1 R. Penrose introduced them for the purposes of quantization of the geometry of space. See [8] for a historical account.

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“…This is obvious for reflections w ∈ W that change the sign of one positive root and permute the others, and it follows in general because W is generated by such reflections, see [BD,V. (4.6) Corollary and (4.10) Lemma].…”

Section: This Impliesmentioning

confidence: 99%

“…Γ ∞ , that are isomorphic to V form the χ-isotypical subspace H χ s ⊂ H s , resp. Γ ∞ χ ⊂ Γ ∞ (see [BD,III.5]). Thus H χ = s∈S H χ s is a Hilbert subfield of H and Γ ∞ χ is a C ∞ (S)-module of its sections.…”

Section: Continuous Sections Let Us Say That a Section ϕ Of H Or K Imentioning

confidence: 99%

Direct Images, Fields of Hilbert Spaces, and Geometric Quantization

Lempert

Szőke

2014

Commun. Math. Phys.

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Abstract. Geometric quantization often produces not one Hilbert space to represent the quantum states of a classical system but a whole family H s of Hilbert spaces, and the question arises if the spaces H s are canonically isomorphic. [ADW] and [Hi] suggest to view H s as fibers of a Hilbert bundle H, introduce a connection on H, and use parallel transport to identify different fibers. Here we explore to what extent this can be done. First we introduce the notion of smooth and analytic fields of Hilbert spaces, and prove that if an analytic field over a simply connected base is flat, then it corresponds to a Hermitian Hilbert bundle with a flat connection and path independent parallel transport. Second we address a general direct image problem in complex geometry: pushing forward a Hermitian holomorphic vector bundle E → Y along a non-proper map Y → S. We give criteria for the direct image to be a smooth field of Hilbert spaces. Third we consider quantizing an analytic Riemannian manifold M by endowing T M with the family of adapted Kähler structures from [LSz2]. This leads to a direct image problem. When M is homogeneous, we prove the direct image is an analytic field of Hilbert spaces. For certain such M-but not all-the direct image is even flat; which means that in those cases quantization is unique. Introduction.This paper is motivated by a problem in geometric quantization: that of uniqueness. At its simplest, geometric quantization is about associating with a Riemannian manifold M a Hermitian line bundle L → X and a Hilbert space H of its sections. In Kähler quantization, L is a holomorphic Hermitian line bundle and H consists of all square integrable holomorphic sections of L. One often knows how to find L, except that its construction involves choices, so that one really has to deal with a family L s → X s of line bundles and Hilbert spaces H s , parametrized by the possible choices s ∈ S. The problem of uniqueness is to find canonical unitary maps H s → H t corresponding to different choices s = t-or rather projective unitary maps, the natural class of maps, since only the projectivized Hilbert spaces have a physical meaning. There are various solutions to this problem, the first the Stone-von Neumann theorem [St1,vN1], long predating geometric quantization. It applies whenever two Hilbert spaces carry irreducible representations of the canonical commutation relations; if so, there is a unitary map, unique up to a scalar factor, that intertwines the two representations. However, the Hilbert spaces that geometric quantization supplies do not carry such representations unless the manifold to be quantized is an affine space. In geometric quantization there is the Blattner-Kostant-Sternberg pairing [Bl1-2,Ko2], which sometimes gives rise to the sought for unitary map, but even in simple cases it may fail to do so [R].In the early 1990s Hitchin in [Hi] and Axelrod, Della Pietra, and Witten in [ADW] considered a situation when the possible choices s form a complex manifold S. (One has to be careful with ...

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Representations of Compact Lie Groups

Cited by 998 publications

References 0 publications

Numerical Continuation of Hamiltonian Relative Periodic Orbits

Numerical Continuation of Hamiltonian Relative Periodic Orbits

Wilson loops in the light of spin networks

Direct Images, Fields of Hilbert Spaces, and Geometric Quantization

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