Characterization of SU q (ℓ + 1)-equivariant spectral triples for the odd dimensional quantum spheres

Chakraborty, Partha Sarathi; Pal, Arupkumar

doi:10.1515/crelle.2008.071

Cited by 17 publications

(41 citation statements)

References 23 publications

(39 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we show that the equivariant spectral triple on S 2 +1 q constructed in [4] has the topological weak heat kernel asymptotic expansion. First let us recall that the odd-dimensional quantum spheres can be realised as the quantum homogeneous space.…”

Section: The Equivariant Spectral Triple On Odd-dimensional Quantum Smentioning

confidence: 94%

“…We denote the representation of C(S 2 +1 q ) on L 2 (S 2 +1 q ) by π eq . In [4] SU q ( + 1) equivariant spectral triples for this covariant representation were studied and a non-trivial one was constructed. It is proved in [14] that the Hilbert space L 2 (S 2 +1 q ) is unitarily equivalent to 2 (N × Z × N ).…”

Section: The Equivariant Spectral Triple On Odd-dimensional Quantum Smentioning

confidence: 99%

“…It is proved in [14] that the Hilbert space L 2 (S 2 +1 q ) is unitarily equivalent to 2 (N × Z × N ). Then the selfadjoint operator D eq constructed in [4] is given on the orthonormal basis {e γ : γ ∈ N × Z × N } by the formula D eq (e γ ) := d γ e γ where d γ is given by…”

Section: The Equivariant Spectral Triple On Odd-dimensional Quantum Smentioning

confidence: 99%

“…Also(1)and (2) are obvious. Now let us now prove(4).We denote the unbounded derivation [|D 0 |, . ],[|D|, .]…”

mentioning

confidence: 99%

See 3 more Smart Citations

Quantum double suspension and spectral triples

Chakraborty

Sundar

2011

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

In this paper we are concerned with the construction of a general principle that will allow us to produce regular spectral triples with finite and simple dimension spectrum. We introduce the notion of weak heat kernel asymptotic expansion (WHKAE) property of a spectral triple and show that the weak heat kernel asymptotic expansion allows one to conclude that the spectral triple is regular with finite simple dimension spectrum. The usual heat kernel expansion implies this property. The notion of quantum double suspension of a C * -algebra was introduced by Hong and Szymanski. Here we introduce the quantum double suspension of a spectral triple and show that the WHKAE is stable under quantum double suspension. Therefore quantum double suspending compact Riemannian spin manifolds iteratively we get many examples of regular spectral triples with finite simple dimension spectrum. This covers all the odd-dimensional quantum spheres. Our methods also apply to the case of noncommutative torus.

show abstract

Section: The Equivariant Spectral Triple On Odd-dimensional Quantum Smentioning

confidence: 94%

Section: The Equivariant Spectral Triple On Odd-dimensional Quantum Smentioning

confidence: 99%

Section: The Equivariant Spectral Triple On Odd-dimensional Quantum Smentioning

confidence: 99%

“…Also(1)and (2) are obvious. Now let us now prove(4).We denote the unbounded derivation [|D 0 |, . ],[|D|, .]…”

mentioning

confidence: 99%

See 2 more Smart Citations

Quantum double suspension and spectral triples

Chakraborty

Sundar

2011

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…We will use the method described in [5] and used implicitly in [3] and [4]. Observe that the self-adjoint operator D in a spectral triple comes with two very crucial restrictions on it, namely, it has to have compact resolvent, and must have bounded commutators with algebra elements.…”

Section: Introductionmentioning

confidence: 99%

Torus Equivariant Spectral Triples for Odd-Dimensional Quantum Spheres Coming from C *-Extensions

Chakraborty

Pal

2007

Lett Math Phys

Self Cite

View full text Add to dashboard Cite

The torus group (S 1 ) ℓ+1 has a canonical action on the odd dimensional sphere S 2ℓ+1 q . We take the natural Hilbert space representation where this action is implemented and characterize all odd spectral triples acting on that space and equivariant with respect to that action. This characterization gives a construction of an optimum family of equivariant spectral triples having nontrivial K-homology class thus generalizing our earlier results for SU q (2). We also relate the triple we construct with the C * -extension

show abstract

Spectral Metric Spaces on Extensions of C*-Algebras

Hawkins¹,

Zacharias

2017

Commun. Math. Phys.

View full text Add to dashboard Cite

Abstract:We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones. Using our construction we find new spectral triples on the quantum 2-and 3-spheres giving a new perspective on these algebras in noncommutative geometry.

show abstract

Characterization of SU q (ℓ + 1)-equivariant spectral triples for the odd dimensional quantum spheres

Abstract: The quantum group SU q (ℓ + 1) has a canonical action on the odd dimensional sphere S

Cited by 17 publications

References 23 publications

Quantum double suspension and spectral triples

Quantum double suspension and spectral triples

Torus Equivariant Spectral Triples for Odd-Dimensional Quantum Spheres Coming from C *-Extensions

Spectral Metric Spaces on Extensions of C*-Algebras

Contact Info

Product

Resources

About