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.
Highlights
In [HL36] Hardy and Littlewood proved the following generalisation of the Plancherel’s identity on the circle T, namely (1 + |m|)p−2| f (m)|p ≤ C p f p L p (T) 1 < p ≤ 2. m∈Z (1.1)The first and third authors were supported in parts by the EPSRC Grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151
We present a version of a Marcinkiewicz interpolation theorem for linear mappings between compact quantum group G of Kac type and the space of matrix-valued sequences that will be realised via
We show that every linear operator A : CD∞ → CD∞ continuous with respect to the Frechet topology can be associated with a distribution K A ‘acting’ on G × G
Summary
In [HL36] Hardy and Littlewood proved the following generalisation of the Plancherel’s identity on the circle T, namely (1 + |m|)p−2| f (m)|p ≤ C p f p L p. It is known that q-deformed quantum groups and homogeneous spaces do not fit into Connes axiomatic framework [Con95] if one wants to have the correct classical limit and various authors have considered modification of the axioms Another problem is that of ‘geometric realisation’ where a given spectral triple operator (understood broadly) should ideally have an interpretation as built from a spin connection and Clifford action on a spinor bundle. We analyse in Theorem 5.3 when D defined by {λπ } is an actual spectral triple in the case of compact matrix quantum group in the sense of Woronowicz [Wor87]. We want to express our gratitude to the anonymous referees for their helpful suggestions
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