Commutator representations of differential calculi on the quantum group SUq(2)

Abstract

Let (Γ, d) be the 3D-calculus or the 4D±-calculus on the quantum group SUq(2). We describe all pairs (π, F ) of a * -representation π of O(SUq (2)) and of a symmetric operator F on the representation space satisfying a technical condition concerning its domain such that there exist a homomorphism of first order differential calculi which maps dx into the commutator [iF, π(x)] for x ∈ O(SUq(2)). As an application commmutator representations of the 2-dimensional leftcovariant calculus on Podles quantum 2-sphere S 2 qc with c = 0 are given.

Spectral Action on SU q (2)

“…The link between π ± and π ± which explains the notations about these intermediate objects and the fact that π ± are representations on different Hilbert spaces, is in the parallel between equations (30), (35) and (36). Let us give immediately a few properties (x β equals x if the sign β is positive and equals x * otherwise)…”

“…5 Differential calculus on SU q (2) and applications 5.1 The sign of D There are multiple differential calculi on SU q (2), see [33,39]. Thanks to [36,Theorem 3], the 3D and 4D ± differential calculi do not coincide with the one considers here: the right multiplication of one-forms by an element in the algebra A is a consequence of the chosen Dirac operator which was introduced according to some equivariance properties with respect to the duality between the two Hopf algebras SU q (2) and U q (su (2)). It is known that the Fredholm module associated to (A, H, D) is one-summable since [F, π(x)] is trace-class for all x ∈ A.…”

Spectral triples and differential calculi related to the Kronecker foliation

“…From the mathematical point of view, only a few types of noncommutative spaces have been used in these examples: commutative algebras of smooth functions on a manifold [1], finite dimensional algebras (for a classification of spectral triples in this case see [6] and [7]) and products of both. In [8] it was shown that it is not straightforward to define spectral triples related to covariant differential calculi on quantum groups. Explicit examples of spectral triples have also been described for the irrational rotation algebra and higher dimensional noncommutative tori [1], [9].…”

Differential calculi over quantum groups

“…We shall sketch a proof in one special case, since the details are rather simple there. For a fuller discussion of "no-go" results along these lines we refer the reader to the papers [9,10] of K. Schmüdgen.…”