164 research outputs found
The K-theory of free quantum groups
In this paper we study the K -theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are K -amenable and establish an analogue of the Pimsner–Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the K -theory of free quantum groups. Our approach relies on a generalization of methods from the Baum–Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum–Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a γ -element and that γ=1 . As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum–Connes conjecture to our setting
Noncommutative elliptic theory. Examples
We study differential operators, whose coefficients define noncommutative
algebras. As algebra of coefficients, we consider crossed products,
corresponding to action of a discrete group on a smooth manifold. We give index
formulas for Euler, signature and Dirac operators twisted by projections over
the crossed product. Index of Connes operators on the noncommutative torus is
computed.Comment: 23 pages, 1 figur
Quantum Field Theory on the Noncommutative Plane with Symmetry
We study properties of a scalar quantum field theory on the two-dimensional
noncommutative plane with quantum symmetry. We start from the
consideration of a firstly quantized quantum particle on the noncommutative
plane. Then we define quantum fields depending on noncommutative coordinates
and construct a field theoretical action using the -invariant measure
on the noncommutative plane. With the help of the partial wave decomposition we
show that this quantum field theory can be considered as a second quantization
of the particle theory on the noncommutative plane and that this field theory
has (contrary to the common belief) even more severe ultraviolet divergences
than its counterpart on the usual commutative plane. Finally, we introduce the
symmetry transformations of physical states on noncommutative spaces and
discuss them in detail for the case of the quantum group.Comment: LaTeX, 26 page
The Hopf modules category and the Hopf equation
We study the Hopf equation which is equivalent to the pentagonal equation,
from operator algebras. A FRT type theorem is given and new types of quantum
groups are constructed. The key role is played now by the classical Hopf
modules category. As an application, a five dimensional noncommutative
noncocommutative bialgebra is given.Comment: 30 pages, Letax2e, Comm. Algebra in pres
A Characterization of right coideals of quotient type and its application to classification of Poisson boundaries
Let be a co-amenable compact quantum group. We show that a right coideal
of is of quotient type if and only if it is the range of a conditional
expectation preserving the Haar state and is globally invariant under the left
action of the dual discrete quantum group. We apply this result to theory of
Poisson boundaries introduced by Izumi for discrete quantum groups and
generalize a work of Izumi-Neshveyev-Tuset on for co-amenable compact
quantum groups with the commutative fusion rules. More precisely, we prove that
the Poisson integral is an isomorphism between the Poisson boundary and the
right coideal of quotient type by maximal quantum subgroup of Kac type. In
particular, the Poisson boundary and the quantum flag manifold are isomorphic
for any q-deformed classical compact Lie group.Comment: 28 pages, Remark 4.9 adde
Groupoids and an index theorem for conical pseudo-manifolds
We define an analytical index map and a topological index map for conical
pseudomanifolds. These constructions generalize the analogous constructions
used by Atiyah and Singer in the proof of their topological index theorem for a
smooth, compact manifold . A main ingredient is a non-commutative algebra
that plays in our setting the role of . We prove a Thom isomorphism
between non-commutative algebras which gives a new example of wrong way
functoriality in -theory. We then give a new proof of the Atiyah-Singer
index theorem using deformation groupoids and show how it generalizes to
conical pseudomanifolds. We thus prove a topological index theorem for conical
pseudomanifolds
Classification of minimal actions of a compact Kac algebra with amenable dual
We show the uniqueness of minimal actions of a compact Kac algebra with
amenable dual on the AFD factor of type II. This particularly implies the
uniqueness of minimal actions of a compact group. Our main tools are a Rohlin
type theorem, the 2-cohomology vanishing theorem, and the Evans-Kishimoto type
intertwining argument.Comment: 68 pages, Introduction rewritten; minor correction
Projective Fourier Duality and Weyl Quantization
The Weyl-Wigner correspondence prescription, which makes large use of Fourier
duality, is reexamined from the point of view of Kac algebras, the most general
background for noncommutative Fourier analysis allowing for that property. It
is shown how the standard Kac structure has to be extended in order to
accommodate the physical requirements. An Abelian and a symmetric projective
Kac algebras are shown to provide, in close parallel to the standard case, a
new dual framework and a well-defined notion of projective Fourier duality for
the group of translations on the plane. The Weyl formula arises naturally as an
irreducible component of the duality mapping between these projective algebras.Comment: LaTeX 2.09 with NFSS or AMSLaTeX 1.1. 102Kb, 44 pages, no figures.
requires subeqnarray.sty, amssymb.sty, amsfonts.sty. Final version with text
improvements and crucial typos correction
Free q-Schrodinger Equation from Homogeneous Spaces of the 2-dim Euclidean Quantum Group
After a preliminary review of the definition and the general properties of
the homogeneous spaces of quantum groups, the quantum hyperboloid qH and the
quantum plane qP are determined as homogeneous spaces of Fq(E(2)). The
canonical action of Eq(2) is used to define a natural q-analog of the free
Schro"dinger equation, that is studied in the momentum and angular momentum
bases. In the first case the eigenfunctions are factorized in terms of products
of two q-exponentials. In the second case we determine the eigenstates of the
unitary representation, which, in the qP case, are given in terms of Hahn-Exton
functions. Introducing the universal T-matrix for Eq(2) we prove that the
Hahn-Exton as well as Jackson q-Bessel functions are also obtained as matrix
elements of T, thus giving the correct extension to quantum groups of well
known methods in harmonic analysis.Comment: 19 pages, plain tex, revised version with added materia
Closed quantum subgroups of locally compact quantum groups
We investigate the fundamental concept of a closed quantum subgroup of a
locally compact quantum group. Two definitions - one due to S.Vaes and one due
to S.L.Woronowicz - are analyzed and relations between them discussed. Among
many reformulations we prove that the former definition can be phrased in terms
of quasi-equivalence of representations of quantum groups while the latter can
be related to an old definition of Podle\'s from the theory of compact quantum
groups. The cases of classical groups, duals of classical groups, compact and
discrete quantum groups are singled out and equivalence of the two definitions
is proved in the relevant context. A deep relationship with the quantum group
generalization of Herz restriction theorem from classical harmonic analysis is
also established, in particular, in the course of our analysis we give a new
proof of Herz restriction theorem.Comment: 24 pages, v3 adds another reference. The paper will appear in
Advances in Mathematic
- …