60 research outputs found
Solutions of Klein--Gordon and Dirac equations on quantum Minkowski spaces
Covariant differential calculi and exterior algebras on quantum homogeneous
spaces endowed with the action of inhomogeneous quantum groups are classified.
In the case of quantum Minkowski spaces they have the same dimensions as in the
classical case. Formal solutions of the corresponding Klein--Gordon and Dirac
equations are found. The Fock space construction is sketched.Comment: 21 pages, LaTeX file, minor change
Global Symmetries of Noncommutative Space-time
The global counterpart of infinitesimal symmetries of noncommutative
space-time is discussed.Comment: 7 pages, no figures; minor changes in the bibliography; final version
accepted for publication in Phys. Rev.
Quasitriangularity and enveloping algebras for inhomogeneous quantum groups
Coquasitriangular universal matrices on quantum Lorentz and
quantum Poincar\'e groups are classified. The results extend (under certain
assumptions) to inhomogeneous quantum groups of [10]. Enveloping algebras on
those objects are described.Comment: 18 pages, LaTeX file, minor change
The Dirac operator and gamma matrices for quantum Minkowski spaces
Gamma matrices for quantum Minkowski spaces are found. The invariance of the
corresponding Dirac operator is proven. We introduce momenta for spin 1/2
particles and get (in certain cases) formal solutions of the Dirac equation.Comment: 25 pages, LaTeX fil
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-deformation of
We construct the action of the quantum double of \uq on the standard
Podle\'s sphere and interpret it as the quantum projective formula generalizing
to the q-deformed setting the action of the Lorentz group of global conformal
transformations on the ordinary Riemann sphere.Comment: LaTeX, 16 pages, we add a reference where an alternative construction
of the q-Lorentz group action on the Podles sphere is give
Geometry of Quantum Principal Bundles I
A theory of principal bundles possessing quantum structure groups and
classical base manifolds is presented. Structural analysis of such quantum
principal bundles is performed. A differential calculus is constructed,
combining differential forms on the base manifold with an appropriate
differential calculus on the structure quantum group. Relations between the
calculus on the group and the calculus on the bundle are investigated. A
concept of (pseudo)tensoriality is formulated. The formalism of connections is
developed. In particular, operators of horizontal projection, covariant
derivative and curvature are constructed and analyzed. Generalizations of the
first structure equation and of the Bianchi identity are found. Illustrative
examples are presented.Comment: 64 pages, AMS-LaTeX, To appear in CM
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
Compactifications of discrete quantum groups
Given a discrete quantum group A we construct a certain Hopf *-algebra AP
which is a unital *-subalgebra of the multiplier algebra of A. The structure
maps for AP are inherited from M(A) and thus the construction yields a
compactification of A which is analogous to the Bohr compactification of a
locally compact group. This algebra has the expected universal property with
respect to homomorphisms from multiplier Hopf algebras of compact type (and is
therefore unique). This provides an easy proof of the fact that for a discrete
quantum group with an infinite dimensional algebra the multiplier algebra is
never a Hopf algebra
Local Index Formula on the Equatorial Podles Sphere
We discuss spectral properties of the equatorial Podles sphere. As a
preparation we also study the `degenerate' (i.e. ) case (related to the
quantum disk). We consider two different spectral triples: one related to the
Fock representation of the Toeplitz algebra and the isopectral one. After the
identification of the smooth pre--algebra we compute the dimension
spectrum and residues. We check the nontriviality of the (noncommutative) Chern
character of the associated Fredholm modules by computing the pairing with the
fundamental projector of the -algebra (the nontrivial generator of the
-group) as well as the pairing with the -analogue of the Bott
projector. Finally, we show that the local index formula is trivially
satisfied.Comment: 18 pages, no figures; minor correction
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