Smooth dense subalgebras and Fourier multipliers on compact quantum groups

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Title: Smooth dense subalgebras and Fourier multipliers on compact quantum groups
Author(s): Akylzhanov, R
Majid, S
Ruzhansky, M
Item Type: Journal Article
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 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 con- struction 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 sub- algebra C ∞ D . We check explicitly that these conditions hold true on the quantum SU q 2 for both its 3-dimensional and 4-dimensional calculi.
Publication Date: 1-Sep-2018
Date of Acceptance: 3-Jun-2018
ISSN: 0010-3616
Publisher: Springer Verlag
Journal / Book Title: Communications in Mathematical Physics
Copyright Statement: This paper is embargoed until publication. Once published will be available fully open access.
Sponsor/Funder: JSC "International Programmes"
The Leverhulme Trust
Engineering & Physical Science Research Council (EPSRC)
Engineering and Physical Sciences Research Council
Funder's Grant Number: RPG-2017-151
Keywords: math.OA
81R50, 43A22
0105 Mathematical Physics
0206 Quantum Physics
0101 Pure Mathematics
Mathematical Physics
Publication Status: Accepted
Embargo Date: publication subject to indefinite embargo
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences

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