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Smooth dense subalgebras and Fourier multipliers on compact quantum groups

Rauan Akylzhanov, Shahn Majid and Michael Ruzhansky (UGent)
(2018) COMMUNICATIONS IN MATHEMATICAL PHYSICS. 362(3). p.761-799
Author
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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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.
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.
Chicago author-date
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.
Chicago author-date (all authors)
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.
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.
IEEE
[1]
R. Akylzhanov, S. Majid, and M. Ruzhansky, “Smooth dense subalgebras and Fourier multipliers on compact quantum groups,” COMMUNICATIONS IN MATHEMATICAL PHYSICS, vol. 362, no. 3, pp. 761–799, 2018.
@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},
  keywords     = {DIRAC OPERATOR,INEQUALITIES,GEOMETRY,ALGEBRA,SUQ(2),SU(2)},
  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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