Topic
Levi decomposition
About: Levi decomposition is a research topic. Over the lifetime, 83 publications have been published within this topic receiving 1823 citations. The topic is also known as: Levi decomposition.
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1 Jan 2020TL;DR: In this article, the authors present a model of a two-level quantum control system with infinite degrees of freedom and a Cartan-CAMP-Levi Decomposition.
Abstract: QUANTUM MECHANICS States and Operators Observables and Measurement Dynamics of Quantum Systems MODELING OF QUANTUM CONTROL SYSTEMS: EXAMPLES Quantum Theory of Interaction of Particles and Fields Approximations and Modeling: Molecular Systems Spin Dynamics and Control Mathematical Structure of Quantum Control Systems CONTROLLABILITY Lie Algebras and Lie Groups Controllability Test: The Dynamical Lie Algebra Notions of Controllability for the State Pure State Controllability Equivalent State Controllability Equality of Orbits OBSERVABILITY AND STATE DETERMINATION Quantum State Tomography Observability Observability and Methods for State Reconstruction LIE GROUP DECOMPOSITIONS AND CONTROL Decompositions of SU(2) and Control of Two Level Systems Decomposition in Planar Rotations Cartan Decompositions Levi Decomposition Examples of Application of Decompositions to Control OPTIMAL CONTROL OF QUANTUM SYSTEMS Formulation of the Optimal Control Problem The Necessary Conditions of Optimality Example: Optimal Control of a Two Level Quantum System Time Optimal Control of Quantum Systems Numerical Methods for Optimal Control of Quantum Systems MORE TOOLS FOR QUANTUM CONTROL Selective Population Transfer via Frequency Tuning Time Dependent Perturbation Theory Adiabatic Control STIRAP Lyapunov Control of Quantum Systems ANALYSIS OF QUANTUM EVOLUTIONS: ENTANGLEMENT, ENTANGLEMENT MEASURES, AND DYNAMICS Entanglement of Quantum Systems Dynamics of Entanglement Local Equivalence of States APPLICATIONS OF QUANTUM CONTROL AND DYNAMICS Nuclear Magnetic Resonance Experiments Molecular Systems Control Atomic Systems Control: Implementations of Quantum Information Processing with Ion Traps APPENDIX A: POSITIVE AND COMPLETELY POSITIVE MAPS, QUANTUM OPERATIONS, AND GENERALIZED MEASUREMENT THEORY Positive and Completely Positive Maps Quantum Operations and Operator Sum Representation Generalized Measurement Theory APPENDIX B: LAGRANGIAN AND HAMILTONIAN FORMALISM IN CLASSICAL ELECTRODYNAMICS Lagrangian Mechanics Extension of Lagrangian Mechanics to Systems with Infinite Degrees of Freedom Lagrangian and Hamiltonian Mechanics for a System of Interacting Particles and Field APPENDIX C: CARTAN SEMISIMPLICITY CRITERION AND CALCULATION OF THE LEVI DECOMPOSITION The Adjoint Representation Cartan Semisimplicity Criterion Quotient Lie Algebras Calculation of the Levi Subalgebra in the Levi Decomposition Algorithm for the Levi Decomposition APPENDIX D: PROOF OF THE CONTROLLABILITY TEST OF THEOREM 3.2.1 APPENDIX E: THE BAKER-CAMPBELL-HAUSDORFF FORMULA AND SOME EXPONENTIAL FORMULAS APPENDIX F: PROOF OF THEOREM 6.2.1 REFERENCES INDEX Notes and Exercises appear at the end of every chapter.
1,023 citations
TL;DR: In this article, all real Lie algebras of dimension up to 8 that admit a nontrivial Levi decomposition was found, and all of them admit a non-parametric nonparametric decomposition.
Abstract: All real Lie algebras of dimension up to 8 that admit a nontrivial Levi decomposition are found.
104 citations
TL;DR: In this article, the authors define a natural algebra homomorphism from H(G,K, V ) to H(Z,Z ∩ K,V U∩K), and show it is injective and identify its image.
Abstract: Let F be a local field with finite residue field of characteristic p. Let G be a connected reductive group over F and B a minimal parabolic subgroup of G with Levi decomposition B = ZU . Let K be a special parabolic subgroup of G, in good position relative to (Z,U). Fix an absolutely irreducible smooth representation of K on a vector space V over some field C of characteristic p. Writing H(G,K, V ) for the intertwining Hecke algebra of V in G, we define a natural algebra homomorphism from H(G,K, V ) to H(Z,Z ∩ K,V U∩K), we show it is injective and identify its image. We thus generalize work of F. Herzig, who assumed F of characteristic 0, G unramified and K hyperspecial, and took for C an algebraic closure of the prime field Fp. We show that in the general case H(G,K, V ) need not be commutative; that is in contrast with the cases Herzig considers and with the more classical situation where V is trivial and the field of coefficients is the field of complex numbers.
56 citations
TL;DR: The structure theory of Lie algebras is used to classify nonlinear systems according to a Levi decomposition and the solvable and semisimple parts of a certain Lie algebra associated with the system as mentioned in this paper.
Abstract: The structure theory of Lie algebras is used to classify nonlinear systems according to a Levi decomposition and the solvable and semisimple parts of a certain Lie algebra associated with the system. An approximation theory is developed and a new class of chaotic systems is introduced, based on the structure theory of Lie algebras.
37 citations
1 Jan 2006
TL;DR: In this paper, it was shown that any Lie algebra g = snr with filiform radical would degenerate to the Lie algebra snfn, where fn is the standard graded filiform Lie algebra of dimension n = dim r.
Abstract: Let g = s n r be an indecomposable Lie algebra with nontrivial semisimple Levi subalgebra s and nontrivial solvable radical r. In this note it is proved that r cannot be isomorphic to a filiform nilpotent Lie algebra. The proof uses the fact that any Lie algebra g = snr with filiform radical would degenerate (even contract) to the Lie algebra snfn, where fn is the standard graded filiform
Lie algebra of dimension n = dim r. This leads to a contradiction, since no such indecomposable algebra snr with r = fn exists
34 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 5 |
| 2020 | 3 |
| 2019 | 2 |
| 2018 | 3 |
| 2017 | 8 |
| 2016 | 3 |