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The Normalizer of a Lie Group. Applications and Challenges
Version 1
: Received: 20 June 2023 / Approved: 20 June 2023 / Online: 20 June 2023 (10:28:44 CEST)
A peer-reviewed article of this Preprint also exists.
Ayala, V.; Da Silva, A.; Torreblanca, M.L. The Normalizer of a Lie Group: Applications and Challenges. Symmetry 2023, 15, 1483. Ayala, V.; Da Silva, A.; Torreblanca, M.L. The Normalizer of a Lie Group: Applications and Challenges. Symmetry 2023, 15, 1483.
Abstract
Let G be a connected Lie group with Lie algebra g. This review is devoted to studying the fundamental dynamic properties of elements in the normalizer NG of G. Through an algebraic characterization of NG we analyze the different dynamics inside the normalizer. NG contains the well-known left-invariant vector fields and the linear and affine vector fields on G. In any case, we show the shape of the solutions of these ordinary differential equations on G. We give examples in low-dimensional Lie groups. It is worth saying that these dynamics generate the linear and
bilinear control systems on Euclidean spaces and the invariant and linear control systems on Lie groups. Moreover, the Jouan Equivalence Theorem shows how to extend this theory to control systems on manifolds.
Keywords
Lie groups; normalizer; invariant; linear and affine vector fields; algebraic control systems
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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