Smooth dense subalgebras and Fourier multipliers on compact quantum
groups
groups
File(s)
Author(s)
Akylzhanov, Rauan
Majid, Shahn
Ruzhansky, M
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 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∞D. We check explicitly that these conditions hold true on the quantum SU2q for both its 3-dimensional and 4-dimensional calculi.
Date Issued
2018-09-01
Date Acceptance
2018-06-03
Citation
Communications in Mathematical Physics, 2018, 362 (3), pp.761-799
ISSN
0010-3616
Publisher
Springer Verlag
Start Page
761
End Page
799
Journal / Book Title
Communications in Mathematical Physics
Volume
362
Issue
3
Copyright Statement
© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Sponsor
JSC "International Programmes"
The Leverhulme Trust
Engineering & Physical Science Research Council (EPSRC)
Engineering and Physical Sciences Research Council
Grant Number
RPG-2017-151
EP/R003025/1
EP/R003025/1
Subjects
Science & Technology
Physical Sciences
Physics, Mathematical
Physics
DIRAC OPERATOR
INEQUALITIES
ALGEBRA
math.OA
math.OA
math.FA
math.QA
81R50, 43A22
0105 Mathematical Physics
0206 Quantum Physics
0101 Pure Mathematics
Mathematical Physics
Publication Status
Published
Date Publish Online
2018-08-13