1,928 research outputs found
Quantum and Braided Linear Algebra
Quantum matrices are known for every matrix obeying the Quantum
Yang-Baxter Equations. It is also known that these act on `vectors' given by
the corresponding Zamalodchikov algebra. We develop this interpretation in
detail, distinguishing between two forms of this algebra, (vectors) and
(covectors). A(R)\to V(R_{21})\tens V^*(R) is an algebra
homomorphism (i.e. quantum matrices are realized by the tensor product of a
quantum vector with a quantum covector), while the inner product of a quantum
covector with a quantum vector transforms as a scaler. We show that if
and are endowed with the necessary braid statistics then their
braided tensor-product V(R)\und\tens V^*(R) is a realization of the braided
matrices introduced previously, while their inner product leads to an
invariant quantum trace. Introducing braid statistics in this way leads to a
fully covariant quantum (braided) linear algebra. The braided groups obtained
from act on themselves by conjugation in a way impossible for the
quantum groups obtained from .Comment: 27 page
Frequency planning for clustered jointly processed cellular multiple access channel
Owing to limited resources, it is hard to guarantee minimum service levels to all users in conventional cellular systems. Although global cooperation of access points (APs) is considered promising, practical means of enhancing efficiency of cellular systems is by considering distributed or clustered jointly processed APs. The authors present a novel `quality of service (QoS) balancing scheme' to maximise sum rate as well as achieve cell-based fairness for clustered jointly processed cellular multiple access channel (referred to as CC-CMAC). Closed-form cell level QoS balancing function is derived. Maximisation of this function is proved as an NP hard problem. Hence, using power-frequency granularity, a modified genetic algorithm (GA) is proposed. For inter site distance (ISD) <; 500 m, results show that with no fairness considered, the upper bound of the capacity region is achievable. Applying hard fairness restraints on users transmitting in moderately dense AP system, 20% reduction in sum rate contribution increases fairness by upto 10%. The flexible QoS can be applied on a GA-based centralised dynamic frequency planner architecture
A note on quantization operators on Nichols algebra model for Schubert calculus on Weyl groups
We give a description of the (small) quantum cohomology ring of the flag
variety as a certain commutative subalgebra in the tensor product of the
Nichols algebras. Our main result can be considered as a quantum analog of a
result by Y. Bazlov
Some Remarks on Producing Hopf Algebras
We report some observations concerning two well-known approaches to
construction of quantum groups. Thus, starting from a bialgebra of
inhomogeneous type and imposing quadratic, cubic or quartic commutation
relations on a subset of its generators we come, in each case, to a q-deformed
universal enveloping algebra of a certain simple Lie algebra. An interesting
correlation between the order of initial commutation relations and the Cartan
matrix of the resulting algebra is observed. Another example demonstrates that
the bialgebra structure of sl_q(2) can be completely determined by requiring
the q-oscillator algebra to be its covariant comodule, in analogy with Manin's
approach to define SL_q(2) as a symmetry algebra of the bosonic and fermionic
quantum planes.Comment: 6 pages, LATEX, no figures, Contribution to the Proceedings of the
4th Colloquium "Quantum Groups and Integrable Systems" (Prague, June 1995
Physics of Quantum Relativity through a Linear Realization
The idea of quantum relativity as a generalized, or rather deformed, version
of Einstein (special) relativity has been taking shape in recent years.
Following the perspective of deformations, while staying within the framework
of Lie algebra, we implement explicitly a simple linear realization of the
relativity symmetry, and explore systematically the resulting physical
interpretations. Some suggestions we make may sound radical, but are arguably
natural within the context of our formulation. Our work may provide a new
perspective on the subject matter, complementary to the previous approach(es),
and may lead to a better understanding of the physics.Comment: 27 pages in Revtex, no figure; proof-edited version to appear in
Phys.Rev.
Hodge Star as Braided Fourier Transform
We study super-braided Hopf algebras primitively generated by
finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules
over a Hopf algebra which are quotients of the augmentation
ideal under right multiplication and the adjoint coaction. Here
super-bosonisation provides a bicovariant differential
graded algebra on . We introduce providing the maximal
prolongation, while the canonical braided-exterior algebra
provides the Woronowicz exterior calculus. In
this context we introduce a Hodge star operator by super-braided
Fourier transform on and left and right interior products by
braided partial derivatives. Our new approach to the Hodge star (a) differs
from previous approaches in that it is canonically determined by the
differential calculus and (b) differs on key examples, having order 3 in middle
degree on with its 3D calculus and obeying the -Hecke relation
in middle degree on with its 4D
calculus. Our work also provided a Hodge map on quantum plane calculi and a new
starting point for calculi on coquasitriangular Hopf algebras whereby any
subcoalgebra defines a sub braided-Lie algebra and
provides the required data .Comment: 36 pages latex 4 pdf figures; minor revision; added some background
in calculus on quantum plane; improved the intro clarit
Deformed Minkowski spaces: clasification and properties
Using general but simple covariance arguments, we classify the `quantum'
Minkowski spaces for dimensionless deformation parameters. This requires a
previous analysis of the associated Lorentz groups, which reproduces a previous
classification by Woronowicz and Zakrzewski. As a consequence of the unified
analysis presented, we give the commutation properties, the deformed (and
central) length element and the metric tensor for the different spacetime
algebras.Comment: Some comments/misprints have been added/corrected, to appear in
Journal of Physics A (1996
Spectral geometry of -Minkowski space
After recalling Snyder's idea of using vector fields over a smooth manifold
as `coordinates on a noncommutative space', we discuss a two dimensional
toy-model whose `dual' noncommutative coordinates form a Lie algebra: this is
the well known -Minkowski space.
We show how to improve Snyder's idea using the tools of quantum groups and
noncommutative geometry.
We find a natural representation of the coordinate algebra of
-Minkowski as linear operators on an Hilbert space study its `spectral
properties' and discuss how to obtain a Dirac operator for this space.
We describe two Dirac operators. The first is associated with a spectral
triple. We prove that the cyclic integral of M. Dimitrijevic et al. can be
obtained as Dixmier trace associated to this triple. The second Dirac operator
is equivariant for the action of the quantum Euclidean group, but it has
unbounded commutators with the algebra.Comment: 23 pages, expanded versio
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