Bilinear Systems Induced by Proper Lie Group Actions Article Swipe
Gong Cheng
,
Wei Zhang
,
Jr-Shin Li
·
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2203.07483
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2203.07483
In the study of induced bilinear systems, the classical Lie algebra rank condition (LARC) is known to be impractical since it requires computing the rank everywhere. On the other hand, the transitive Lie algebra condition, while more commonly used, relies on the classification of transitive Lie algebras, which is elusive except for few simple geometric objects such as spheres. We prove in this note that for bilinear systems induced by proper Lie group actions, the underlying Lie algebra is closely related to the orbits of the group action. Knowing the pattern of the Lie algebra rank over the manifold, we show that the LARC can be relaxed so that it suffices to check the rank at an arbitrary single point. Moreover, it removes the necessity for classifying transitive Lie algebras. Finally, this relaxed rank condition also leads to a characterization of controllable submanifolds by orbits.
Related Topics
Concepts
Mathematics
Pure mathematics
Lie algebra
Rank (graph theory)
Simple Lie group
Lie group
Lie conformal algebra
Algebra over a field
Adjoint representation of a Lie algebra
Killing form
Transitive relation
Graded Lie algebra
Adjoint representation
Representation of a Lie group
Combinatorics
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2203.07483
- https://arxiv.org/pdf/2203.07483
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4221142338
All OpenAlex metadata
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https://openalex.org/W4221142338Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2203.07483Digital Object Identifier
- Title
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Bilinear Systems Induced by Proper Lie Group ActionsWork title
- Type
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2022Year of publication
- Publication date
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2022-03-14Full publication date if available
- Authors
-
Gong Cheng, Wei Zhang, Jr-Shin LiList of authors in order
- Landing page
-
https://arxiv.org/abs/2203.07483Publisher landing page
- PDF URL
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https://arxiv.org/pdf/2203.07483Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2203.07483Direct OA link when available
- Concepts
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Mathematics, Pure mathematics, Lie algebra, Rank (graph theory), Simple Lie group, Lie group, Lie conformal algebra, Algebra over a field, Adjoint representation of a Lie algebra, Killing form, Transitive relation, Graded Lie algebra, Adjoint representation, Representation of a Lie group, CombinatoricsTop concepts (fields/topics) attached by OpenAlex
- Cited by
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0Total citation count in OpenAlex
- Related works (count)
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10Other works algorithmically related by OpenAlex
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