108 research outputs found
Symmetries of quantum spaces. Subgroups and quotient spaces of quantum and groups
We prove that each action of a compact matrix quantum group on a compact
quantum space can be decomposed into irreducible representations of the group.
We give the formula for the corresponding multiplicities in the case of the
quotient quantum spaces. We describe the subgroups and the quotient spaces of
quantum SU(2) and SO(3) groups.Comment: 30 pages (with very slight changes
q-deformed Dirac Monopole With Arbitrary Charge
We construct the deformed Dirac monopole on the quantum sphere for arbitrary
charge using two different methods and show that it is a quantum principal
bundle in the sense of Brzezinski and Majid. We also give a connection and
calculate the analog of its Chern number by integrating the curvature over
.Comment: Technical modifications made on the definition of the base. A more
geometrical trivialization is used in section
The distribution of geodesic excursions into the neighborhood of a cone singularity on a hyperbolic 2-orbifold
A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an
infinite sequence of penetrations into a neighborhood of a cone singularity, so
that the sequence of depths of maximal penetration has a limiting distribution.
The distribution function is the same for all such surfaces and is described by
a fairly simple formula.Comment: 20 page
On a correspondence between quantum SU(2), quantum E(2) and extended quantum SU(1,1)
In a previous paper, we showed how one can obtain from the action of a
locally compact quantum group on a type I-factor a possibly new locally compact
quantum group. In another paper, we applied this construction method to the
action of quantum SU(2) on the standard Podles sphere to obtain Woronowicz'
quantum E(2). In this paper, we will apply this technique to the action of
quantum SU(2) on the quantum projective plane (whose associated von Neumann
algebra is indeed a type I-factor). The locally compact quantum group which
then comes out at the other side turns out to be the extended SU(1,1) quantum
group, as constructed by Koelink and Kustermans. We also show that there exists
a (non-trivial) quantum groupoid which has at its corners (the duals of) the
three quantum groups mentioned above.Comment: 35 page
Bicovariant Quantum Algebras and Quantum Lie Algebras
A bicovariant calculus of differential operators on a quantum group is
constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given
by elements of the pure braid group. These operators --- the `reflection
matrix' being a special case --- generate algebras that
linearly close under adjoint actions, i.e. they form generalized Lie algebras.
We establish the connection between the Hopf algebra formulation of the
calculus and a formulation in compact matrix form which is quite powerful for
actual computations and as applications we find the quantum determinant and an
orthogonality relation for in .Comment: 38 page
The Problem of Differential Calculus on Quantum Groups
The bicovariant differential calculi on quantum groups of Woronowicz have the
drawback that their dimensions do not agree with that of the corresponding
classical calculus. In this paper we discuss the first-order differential
calculus which arises from a simple quantum Lie algebra. This calculus has the
correct dimension and is shown to be bicovariant and complete. But it does not
satisfy the Leibniz rule. For sl_n this approach leads to a differential
calculus which satisfies a simple generalization of the Leibniz rule.Comment: Contribution to the proceedings of the Colloquium on Quantum Groups
and Integrable Systems Prague, June 1996. amslatex, 9 pages. For related
information see http://www.mth.kcl.ac.uk/~delius/q-lie.htm
Homomorphisms of quantum groups
In this article, we study several equivalent notions of homomorphism between locally compact quantum groups compatible with duality. In particular, we show that our homomorphisms are equivalent to functors between the respective categories of coactions. We lift the reduced bicharacter to universal quantum groups for any locally compact quantum group defined by a modular multiplicative unitary, without assuming Haar weights. We work in the general setting of modular multiplicative unitaries
Quantum teardrops
Algebras of functions on quantum weighted projective spaces are introduced,
and the structure of quantum weighted projective lines or quantum teardrops are
described in detail. In particular the presentation of the coordinate algebra
of the quantum teardrop in terms of generators and relations and classification
of irreducible *-representations are derived. The algebras are then analysed
from the point of view of Hopf-Galois theory or the theory of quantum principal
bundles. Fredholm modules and associated traces are constructed. C*-algebras of
continuous functions on quantum weighted projective lines are described and
their K-groups computed.Comment: 18 page
Quantum isometries and noncommutative spheres
We introduce and study two new examples of noncommutative spheres: the
half-liberated sphere, and the free sphere. Together with the usual sphere,
these two spheres have the property that the corresponding quantum isometry
group is "easy", in the representation theory sense. We present as well some
general comments on the axiomatization problem, and on the "untwisted" and
"non-easy" case.Comment: 16 page
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