Structure and Geometry of Lie Groups

Hilgert, Joachim; Neeb, Karl-Hermann

doi:10.1007/978-0-387-84794-8

Cited by 267 publications

(212 citation statements)

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“…Such a map is constructed as follows: let us recall, e.g. from [31], that every element γ ∈ Sp(2n, R) can be decomposed as…”

Section: Dynamical Decoupling For Quadratic Hamiltoniansmentioning

confidence: 99%

Dynamical decoupling and homogenization of continuous variable systems

Arenz¹,

Burgarth²,

Hillier³

2017

J. Phys. A: Math. Theor.

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Abstract. For finite-dimensional quantum systems, such as qubits, a well established strategy to protect such systems from decoherence is dynamical decoupling. However many promising quantum devices, such as oscillators, are infinite dimensional, for which the question if dynamical decoupling could be applied remained open. Here we first show that not every infinite-dimensional system can be protected from decoherence through dynamical decoupling. Then we develop dynamical decoupling for continuous variable systems which are described by quadratic Hamiltonians. We identify a condition and a set of operations that allow us to map a set of interacting harmonic oscillators onto a set of non-interacting oscillators rotating with an averaged frequency, a procedure we call homogenization. Furthermore we show that every quadratic system-environment interaction can be suppressed with two simple operations acting only on the system. Using a random dynamical decoupling or homogenization scheme, we develop bounds that characterize how fast we have to work in order to achieve the desired uncoupled dynamics. This allows us to identify how well homogenization can be achieved and decoherence can be suppressed in continuous variable systems.

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“…Such a map is constructed as follows: let us recall, e.g. from [31], that every element γ ∈ Sp(2n, R) can be decomposed as…”

Section: Dynamical Decoupling For Quadratic Hamiltoniansmentioning

confidence: 99%

Dynamical decoupling and homogenization of continuous variable systems

Arenz¹,

Burgarth²,

Hillier³

2017

J. Phys. A: Math. Theor.

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“…In this paper the concept of Lie group and Lie algebra is used extensively, [28]. This section gives selected details concerning description of control systems on Lie groups.…”

Section: Lie Algebra and Lie Groupmentioning

confidence: 99%

Control of a planar robot in the flight phase using transverse function approach

Pazderski¹,

Kozłowski²

2015

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. This paper deals with stabilization and tracking control problems defined with respect to a planar mechanical structure similar to Raibert's robot. The proposed control solution is based on formal analysis of the control system on a Lie group. In order to take advantage of Lie group theory a dynamic extension of the robot kinematics is introduced. To cope with non-zero angular momentum the controller based on transverse functions is employed. Properties of the closed-loop control system are investigated based on simulations including practical stabilization at neighborhood of a constant point or a reference trajectory.

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“…Remark 8. 22. In this remark, we want to point out that there exist nontrivial algebra bundles which are trivial as noncommutative principal torus bundles: In fact, [15,Corollary 12.7] implies that the 2-tori A 2 θ are mutually nonisomorphic for 0 θ 1 2 .…”

Section: Example 2: Sections Of Algebra Bundles With Trivial Noncommumentioning

confidence: 99%

“…It is a well-known fact that SU 3 (C) is a 1-connected, that is, connected and simply connected, Lie group (cf. [22,Proposition 16.1.3]). Moreover, SU 3 (C) contains a maximal torus of dimension 2.…”

Section: Example 3: Sections Of Algebra Bundles Which Are Pull-backs mentioning

confidence: 99%

A geometric approach to noncommutative principal torus bundles

Wagner

2013

Proceedings of the London Mathematical Society

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A (smooth) dynamical system with transformation group T n is a triple (A, T n , α), consisting of a unital locally convex algebra A, the n-torus T n and a group homomorphism α : T n → Aut(A), which induces a (smooth) continuous action of T n on A. In this paper, we present a new, geometrically oriented approach to the noncommutative geometry of principal torus bundles based on such dynamical systems. Our approach is inspired by the classical setting: In fact, after recalling the definition of a trivial noncommutative principal torus bundle, we introduce a convenient (smooth) localization method for noncommutative algebras and say that a dynamical system (A, T n , α) is called a noncommutative principal T n -bundle, if localization leads to a trivial noncommutative principal T n -bundle. We prove that this approach extends the classical theory of principal torus bundles and present a bunch of (nontrivial) noncommutative examples. Contents

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Structure and Geometry of Lie Groups

Cited by 267 publications

References 0 publications

Dynamical decoupling and homogenization of continuous variable systems

Dynamical decoupling and homogenization of continuous variable systems

Control of a planar robot in the flight phase using transverse function approach

A geometric approach to noncommutative principal torus bundles

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