Smooth dense subalgebras and Fourier multipliers on compact quantum groups
- Author
- Rauan Akylzhanov, Shahn Majid and Michael Ruzhansky (UGent)
- Organization
- Abstract
- We define and study dense Frechet subalgebras of compact quantum groups realised as smooth domains associated with a Dirac type operator with compact resolvent. Further, we construct spectral triples on compact matrix quantum groups in terms of Clebsch-Gordon coefficients and the eigenvalues of the Dirac operator . Grotendieck's theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on them. It is also shown that regular pseudo-differential operators are closed under compositions. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for L (p) - L (q) boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to our proposed smooth subalgebra . We check explicitly that these conditions hold true on the quantum SU2 (q) for both its 3-dimensional and 4-dimensional calculi.
- Keywords
- DIRAC OPERATOR, INEQUALITIES, GEOMETRY, ALGEBRA, SUQ(2), SU(2)
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU-8585464
- Chicago
- Akylzhanov, Rauan, Shahn Majid, and Michael Ruzhansky. 2018. “Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups.” Communications in Mathematical Physics 362 (3): 761–799.
- APA
- Akylzhanov, Rauan, Majid, S., & Ruzhansky, M. (2018). Smooth dense subalgebras and Fourier multipliers on compact quantum groups. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 362(3), 761–799.
- Vancouver
- 1.Akylzhanov R, Majid S, Ruzhansky M. Smooth dense subalgebras and Fourier multipliers on compact quantum groups. COMMUNICATIONS IN MATHEMATICAL PHYSICS. 2018;362(3):761–99.
- MLA
- Akylzhanov, Rauan, Shahn Majid, and Michael Ruzhansky. “Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups.” COMMUNICATIONS IN MATHEMATICAL PHYSICS 362.3 (2018): 761–799. Print.
@article{8585464,
abstract = {We define and study dense Frechet subalgebras of compact quantum groups realised as smooth domains associated with a Dirac type operator with compact resolvent. Further, we construct spectral triples on compact matrix quantum groups in terms of Clebsch-Gordon coefficients and the eigenvalues of the Dirac operator . Grotendieck's theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on them. It is also shown that regular pseudo-differential operators are closed under compositions. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for L (p) - L (q) boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to our proposed smooth subalgebra . We check explicitly that these conditions hold true on the quantum SU2 (q) for both its 3-dimensional and 4-dimensional calculi.},
author = {Akylzhanov, Rauan and Majid, Shahn and Ruzhansky, Michael},
issn = {0010-3616},
journal = {COMMUNICATIONS IN MATHEMATICAL PHYSICS},
language = {eng},
number = {3},
pages = {761--799},
title = {Smooth dense subalgebras and Fourier multipliers on compact quantum groups},
url = {http://dx.doi.org/10.1007/s00220-018-3219-4},
volume = {362},
year = {2018},
}
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