Quantum Dimension and Quantum Projective Spaces

Abstract

Abstract. We show that the family of spectral triples for quantum projective spaces introduced by D'Andrea and Dabrowski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action of the element K 2ρ or its inverse. The spectral dimension computed in this sense coincides with the dimension of the classical projective spaces. The connection with the well known notion of quantum dimension of quantum group theory is pointed out.

An analogue of Weyl’s law for quantized irreducible generalized flag manifolds

“…Notice that for the other cases the formula is more complicated, since max{a Remark 4. A computation of the spectral dimension for quantum projective spaces was given in [Mat14]. It can be checked that the Casimir operator used in this reference corresponds, in the language of the present paper, to the choice of the fundamental representation and t = 1/2.…”

“…As discussed in [Mat14], the lack of the trace property for the Haar state must be accounted for in the residue formula. This means introducing explicitely the modular operator implementing the automorphism θ, which we denote by ∆ h .…”

Crossed product extensions of spectral triples

“…33),Tr Θ π(f ) (1 + D 2 ) −s/2 π(g) = 2 n∈Z e −n Tr H (g ⋆ β(f ))(0)[1 + D 2 + (η + ω e −n ) 2 ] −s/2 = 2 ϕ R (g ⋆ β(f )) n∈Z R db e −n [1 + b 2 + (η + ω e −n ) 2 ] −s/2 = 2 ϕ R (g ⋆ β(f )) √ π Γ((s−1)/2) Γ(s/2) n∈Z e −n [1 + (η + ω e −n ) 2 ] (1−s)/2 . Let a n (s) := e −n [1 + (η + ω e −n ) 2 ] (1−s)/2 .…”