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Definitions and Concepts
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
Lie algebra A Lie algebra is a vector space equipped with a Lie bracket (often called a commutator) [x,y] that satisfies three axioms (see Olver [942]): [x,y] is bilinear (i.e., linear in each of x and y separately),the Lie bracket is anti-commutative (i.e., [x,y]=−[y,x]),the Jacobi identity, [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0, holds.
Introduction
Published in Sabiha Wadoo, Pushkin Kachroo, Autonomous Underwater Vehicles, 2017
Given two vector fields, f and g, a new vector field can be defined by ([f,g](p))λ=(LfLgλ)(p)−(LgLfλ)(p) and [f,g] is called the Lie bracket of f and g. A vector space with the Lie bracket as the binary operation forms a Lie algebra. Denote by V∞(M) the space of C∞ vector fields on M.
Symmetries and Group Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
and it follows that [J1, J2] = 0. The Lie bracket is the algebraic structure in the Lie algebra and specifying how it acts on the different generators uniquely defines the local behaviour of the group. Furthermore, since the Lie bracket of two elements results in a new element close to the identity, the meaning of the summation should be clearer as we are adding elements of the Lie algebra, which is a vector space, together. For group elements that are further away from the identity, we would need to include higher orders of the expansion, but they will all include Lie brackets and result in adding elements of the Lie algebra together.
Modeling of DC–DC Converter Using Exact Feedback Linearization Method: A Discussion
Published in IETE Journal of Research, 2019
Debanjana Bhattacharyya, Subhransu Padhee, Kishor Chandra Pati
EFL method of linearization is based on Lie algebra and differential geometry. Lie algebra is defined as a vector space L over a field F with the operation, Lie bracket (x, y)↦[x, y] and this vector space L satisfies the following properties [30]. The operation of lie bracket is bilinear[x, x] = 0 for all x ∈ LJacobi identity which is represented as [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 , ∀x, y, z ∈ L
Stability Analysis of DC-DC Converters Applying Lie Algebra
Published in IETE Journal of Research, 2022
Debanjana Bhattacharyya, Kishor Chandra Pati
A Lie Algebra L is defined as a vector space over a field F with an operation called Lie bracket . This vector space complies with the following properties [31]. The Lie bracket operation is bilinear.Jacobi identity which is given as