76 research outputs found
From C*algebra extensions to CQMS, , Podles sphere and other examples
We construct compact quantum metric spaces (CQMS) starting with some
C*algebra extension with a positive splitting. As special cases we discuss the
case of Toeplitz algebra, quantum SU(2) and Podles sphere
Metrics On The Quantum Heisenberg Manifold
Compact quantum metric spaces are order unit spaces along with a Lip norm. On
the order unit space of the selfadjoint elements of the dense subalgebra of
smooth elements in the quantum Heisenberg manifold we construct Lip norms
Connes' calculus for The Quantum double suspension
Given a spectral triple Connes associated a
canonical differential graded algebra .
However, so far this has been computed for very few special cases. We identify
suitable hypotheses on a spectral triple that helps one to compute the
associated Connes' calculus for its quantum double suspension. This allows one
to compute for spectral triples obtained by iterated
quatum double suspension of the spectral triple associated with a first order
differential operator on a compact smooth manifold. This gives the first
systematic computation of Connes' calculus for a large family of spectral
triples
Gelfand-Kirillov dimension of the algebra of regular functions on quantum groups
Let be the -deformation of a simply connected simple compact Lie
group of type , or and be the algebra of
regular functions on . In this article, we prove that the Gelfand-Kirillov
dimension of is equal to the dimension of real manifold .Comment: 15 pages. arXiv admin note: text overlap with arXiv:1709.0858
Characterization of -equivariant spectral triples for the odd dimensional quantum spheres
The quantum group has a canonical action on the odd
dimensional sphere . All odd spectral triples acting on the
space of and equivariant under this action have been
characterized. This characterization then leads to the construction of an
optimum family of equivariant spectral triples having nontrivial -homology
class. These generalize the results of Chakraborty & Pal for .Comment: LaTeX2e, 20 page
Gelfand-Kirillov dimension of some simple unitarizable modules
Let be the quantized algebra of regular functions on a
semisimple simply connected compact Lie group . Simple unitarizable left
-module are classified.
In this article, we compute their Gelfand-Kirillov dimension where is of
the type , and .Comment: 19 page
Local index formula for the quantum double suspension
Our understanding of local index formula in noncommutative geometry is
stalled for a while because we do not have more than one explicit computation,
namely that of Connes for quantum SU(2) and do not understand the meaning of
the various multilinear functionals involved in the formula. In such a
situation further progress in understanding necessitates more explicit
computations and here we execute the second explicit computation for the
quantum double suspension, a construction inspired by the Toeplitz extension.
More specifically we compute local index formula for the quantum double
suspensions of and the noncommutative -torus.Comment: Spelling mistake in the name of second author has been rectifie
K-groups of the quantum homogeneous space
Quantum Steiffel manifolds were introduced by Vainerman and Podkolzin in
\cite{VP}. They classified the irreducible representations of their underlying
-algebras. Here we compute the K groups of the quantum homogeneous spaces
. Specializing to the case we show that
the fundamental unitary for quantum is nontrivial and is a unimodular
element in
Characterization of spectral triples: A combinatorial approach
We describe a general technique to study Dirac operators on noncommutative
spaces under some additional assumptions. The main idea is to capture the
compact resolvent condition in a combinatorial set up. Using this, we then
prove that for a certain class of representations of the C^*-algebra
C(SU_q(\ell+1)), any Dirac operator that diagonalises with respect to the
natural basis of the underlying Hilbert space must have trivial sign.Comment: v3: partly rewritten; the equivariant case has now been taken out and
would be treated in a separate paper. v2: few typos corrected. LaTeX2e, uses
xy-pic and eepi
The weak heat kernel asymptotic expansion and the quantum double suspension
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. Finally we show that WHKAE is stable under quantum
double suspension, a notion introduced by Hong and Szymanski. 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
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