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 \({\mathcal{D}}\). 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 Lp − Lq 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 \({{C}^\infty_\mathcal {D}}\). We check explicitly that these conditions hold true on the quantum SU2q for both its 3-dimensional and 4-dimensional calculi.
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Communicated by C. Schweigert
The first and third authors were supported in parts by the EPSRC Grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151. The first author was also partially supported by the Simons-Foundation Grant 346300 and the Polish Government MNiSW 2015–2019 matching fund. No new data was created or generated during the course of this research.
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Akylzhanov, R., Majid, S. & Ruzhansky, M. Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups. Commun. Math. Phys. 362, 761–799 (2018). https://doi.org/10.1007/s00220-018-3219-4
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DOI: https://doi.org/10.1007/s00220-018-3219-4