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Quantum Filtering in Robotics, Information, Feynman Path Integrals, Levy Noise, Haar Measure on Groups, Gravity and Robots, Canonical Quantum Gravity, Langevin Equation, Antenna Current in a Field Dependent Medium
Published in Harish Parthasarathy, Electromagnetics, Control and Robotics, 2023
In general for any semi-simple Lie algebra g with G = exp (g), we have the root space decomposition g=h⊕⊕α∈Δgα=h⊕⊕α∈Δ+(gα⊕gα)[H,Hα]=α(H)Xα,Xα∈gα,dimgα=0
About the Solution of Some Inverse Problems in Differential Galois Theory by Hamburger Equations
Published in K. D. Elworthy, W. Norrie Evenitt, E. Bruce Lee, Differential equations, dynamical systems, and control science, 2017
For the proof of our main theorem we will make an essential use of a Chevalley basis. Let ℒ be a semi-simple Lie algebra. A Chevalley basis for ℒ is a set {Hi, Xi, Yi}i = 1,…,n of generators satisfying the relations :
Lie Algebras
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Let V be a vector space of dimension n ≥ 1 over a field of characteristic 0. Let W(n) = Der (S(V)). Then W(n) is an infinite-dimensional simple Lie algebra.
Kinematical Lie Algebras and Invariant Functions of Algebras
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2019
J. M. Escobar, J. Núñez, P. Pérez-Fernández
where are the entries of a real symmetric matrix with rows and columns ordered a, b, c and d, called the contraction matrix, which will be denoted by ε. complex simple Lie algebra denoted by B2
High-order contact transformations of molecular Hamiltonians: general approach, fast computational algorithm and convergence of ro-vibrational polyad models
Published in Molecular Physics, 2022
Vladimir Tyuterev, Sergey Tashkun, Michael Rey, Andrei Nikitin
The algebra involving powers of the corresponding rotational operator is known in mathematics as Poincaré enveloping algebra of simple Lie algebra . The Casimir operator J2 of the algebra commute with all other elements. The structural constants of the enveloping rotational algebra are exactly known [39] and are given in Appendix 3 in the symmetrised form. In case of the dipole CTs, the cosine directors are involved, which do not commute with rotational operators The rotational enveloping algebra of CT for the dipole moment is thus significantly larger. A supplementary complication arises because J2 does not play anymore the role of the Casimir operator since it does not commute with the direction cosines . Because of their algebraic properties, the direction cosines are involved as a ‘linear factor’ in the transformed dipole moment where the rotational factors involve total angular moment components in the Eckart MFF frame The vibrational operators keep exactly the same properties both for the commutators (Appendix 2) and for the matrix elements, whereas related calculations for the rotational operators (which are written here in a simplified concise form) are more complicated. In the works by Lamouroux et al. [154,344], the structure constants of the rotational CT algebra for the dipole moment were computed in a specific symmetrised form up to eighth total power of rotational operators.