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A general approach to noncommutative spaces from Poisson homogeneous spaces: Applications to (A)dS and Poincar\'e
by Angel Ballesteros, Ivan Gutierrez-Sagredo and Francisco J. Herranz
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Francisco J. Herranz |
| Submission information | |
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| Preprint Link: | scipost_202212_00061v1 (pdf) |
| Date submitted: | Dec. 22, 2022, 3:15 p.m. |
| Submitted by: | Francisco J. Herranz |
| Submitted to: | SciPost Physics Proceedings |
| Proceedings issue: | 34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
In this contribution we present a general procedure that allows the construction of noncommutative spaces with quantum group invariance as the quantization of their associated coisotropic Poisson homogeneous spaces coming from a coboundary Lie bialgebra structure. The approach is illustrated by obtaining in an explicit form several noncommutative spaces from (3+1)D (A)dS and Poincar\'e coisotropic Lie bialgebras. In particular, we review the construction of the $\kappa$-Minkowski and $\kappa$-(A)dS spacetimes in terms of the cosmological constant $\Lambda$. Furthermore, we present all noncommutative Minkowski and (A)dS spacetimes that preserved a quantum Lorentz subgroup. Finally, it is also shown that the same setting can be used to construct the three possible 6D $\kappa$-Poincar\'e spaces of time-like.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2023-1-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202212_00061v1, delivered 2023-01-24, doi: 10.21468/SciPost.Report.6601
Strengths
- The paper provides a rigorous construction of a number of nontrivial, noncommutative, associative algebras starting from classical, geometric structures based on Lie groups.
- The presentation is concise and clear with all basic ingredients defined carefully and many examples worked out explicitly.
Weaknesses
- The algebras are described only in the defining, or adjoint, representation, whereas it would h have been interesting if the authors could have commented on the existence of other representations, for example by factoring out nontrivial ideals.