Journal of Nonlinear Mathematical Physics

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Volume 25, Issue 4, July 2018, Pages 528 - 557

Composition of Lie Group Elements from Basis Lie Algebra Elements

Authors
George W. Bluman
Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2, Canada,bluman@math.ubc.ca
Omar Mrani-Zentar
Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2, Canada,o.mrani@math.ubc.ca
Deshin Finlay
Department of Mathematics, University of Michigan, Ann Arbor 48109-1043, USA,dfinlay@umich.edu
Received 15 June 2017, Accepted 27 April 2018, Available Online 6 January 2021.
DOI
10.1080/14029251.2018.1503398How to use a DOI?
Keywords
Lie groups; Lie algebras; commutators
Abstract

It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number of ways with other movements generated by compositions of allowed motions. Two concrete examples are: (1) the restricted parallel parking problem where the commutator of translations in y and rotations in the xy-plane yields translations in x. Here the control problem involves a vehicle that can only perform a series of translations in y and rotations with the aim of efficiently obtaining a pure translation in x; (2) involves an apparatus that can only perform rotations about two axes with the aim of performing rotations about a third axis. Both examples involve three-dimensional Lie algebras. In particular, the composition problem is solved for the nine three- and four-dimensional Lie algebras with non-trivial solutions. Three different solution methods are presented. Two of these methods depend on operator and matrix representations of a Lie algebra. The other method is a differential equation method that depends solely on the commutator properties of a Lie algebra. Remarkably, for these distinguished Lie algebras the solutions involve arbitrary functions and can be expressed in terms of elementary functions.

Copyright
© 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
25 - 4
Pages
528 - 557
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2018.1503398How to use a DOI?
Copyright
© 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

risenwbib
TY  - JOUR
AU  - George W. Bluman
AU  - Omar Mrani-Zentar
AU  - Deshin Finlay
PY  - 2021
DA  - 2021/01/06
TI  - Composition of Lie Group Elements from Basis Lie Algebra Elements
JO  - Journal of Nonlinear Mathematical Physics
SP  - 528
EP  - 557
VL  - 25
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1503398
DO  - 10.1080/14029251.2018.1503398
ID  - Bluman2021
ER  -