3,431 research outputs found
Positive Representations of Split Real Simply-laced Quantum Groups
We construct the positive principal series representations for
where is of
simply-laced type, parametrized by where is the
rank of . We describe explicitly the actions of the generators in
the positive representations as positive essentially self-adjoint operators on
a Hilbert space, and prove the transcendental relations between the generators
of the modular double. We define the modified quantum group
of the
modular double and show that the representations of both parts of the modular
double commute weakly with each other, there is an embedding into a quantum
torus algebra, and the commutant contains its Langlands dual.Comment: Finalized published version. Introduction has been rewritten to
reflect recent progress and references added. Some typos fixe
Positive representations, multiplier Hopf algebra, and continuous canonical basis
We introduce the language of multiplier Hopf algebra in the context of
positive representations of split real quantum groups, and discuss its
applications with a continuous version of Lusztig-Kashiwara's canonical basis,
which may provide a key to prove the closure of the positive representations
under tensor products, and harmonic analysis of quantized algebra of functions
in the sense of locally compact quantum groups.Comment: Revised version for publication to Proceedings of 2013 RIMS
Conference "String theory, integrable systems and representation theory"
Extended Section 2, added Section 3 and 5, updated reference
Gauss-Lusztig Decomposition for and Representation by q-Tori
We found an explicit construction of a representation of the positive quantum
group and its modular double GL_{q\til[q]}^+(N,\R) by positive
essentially self-adjoint operators. Generalizing Lusztig's parametrization, we
found a Gauss type decomposition for the totally positive quantum group
for , parametrized by the standard decomposition of the
longest element . Under this parametrization, we found
explicitly the relations between the standard quantum variables, the relations
between the quantum cluster variables, and realizing them using non-compact
generators of the -tori by positive essentially self-adjoint
operators. The modular double arises naturally from the transcendental
relations, and an L^2(GL_{q\til[q]}^+(N,\R)) space in the von Neumann setting
can also be defined.Comment: Reorganizing the contents involving positivity. Renewed reference
Positive Casimir and Central Characters of Split Real Quantum Groups
We describe the generalized Casimir operators and their actions on the
positive representations of the modular double of split real
quantum groups . We introduce the notion of virtual
highest and lowest weights, and show that the central characters admit positive
values for all parameters . We show that their image defines a
semi-algebraic region bounded by real points of the discriminant variety
independent of , and we discuss explicit examples in the lower rank cases.Comment: 33 pages, 6 figures Expanded introduction. Minor typo fixe
Positive representations of non-simply-laced split real quantum groups
We construct the positive principal series representations for
where is of type , , or , parametrized by where
is the rank of . We show that under the representations, the generators
of the Langlands dual group are related to the
generators of by the transcendental relations. We define the
modified quantum group of the modular double and show that the representations
of both parts of the modular double commute with each other, and there is an
embedding into the -tori polynomials.Comment: Title changed. Fixed typos in the representations of C_n and F_4.
Fixed typos in the matrix at (4.48) and signs of lambda. Add a remark on the
representation of type G
On tensor products of positive representations of split real quantum Borel subalgebra
We studied the positive representations of split real quantum
groups restricted to the Borel subalgebra
. We proved that the restriction is independent of the
parameter . Furthermore, we prove that it can be constructed from the
GNS-representation of the multiplier Hopf algebra
constructed earlier, which enables us to decompose their tensor product using
the theory of the "multiplicative unitary". This will be an essential
ingredient in the construction of quantum higher Teichm\"{u}ller theory from
the perspective of representation theory, generalizing earlier work by
Frenkel-Kim.Comment: Revised version for publication. Introduction is rewritten. Section
6.3 is removed to shorten the pape
Q-operator and fusion relations for
The construction of the Q-operator for twisted affine superalgebra
is given. It is shown that the corresponding prefundamental
representations give rise to evaluation modules some of which do not have a
classical limit, which nevertheless appear to be a necessary part of fusion
relations.Comment: 22 p, published versio
Supersymmetry and the Modular Double
A counterpart of the modular double for quantum superalgebra
\cU_q(\osp(1|2)) is constructed by means of supersymmetric quantum mechanics.
We also construct the -matrix operator acting in the corresponding
representations, which is expressed via quantum dilogarithm.Comment: 21 page
Implementation of a FPGA-Based Feature Detection and Networking System for Real-time Traffic Monitoring
With the growing demand of real-time traffic monitoring nowadays,
software-based image processing can hardly meet the real-time data processing
requirement due to the serial data processing nature. In this paper, the
implementation of a hardware-based feature detection and networking system
prototype for real-time traffic monitoring as well as data transmission is
presented. The hardware architecture of the proposed system is mainly composed
of three parts: data collection, feature detection, and data transmission.
Overall, the presented prototype can tolerate a high data rate of about 60
frames per second. By integrating the feature detection and data transmission
functions, the presented system can be further developed for various VANET
application scenarios to improve road safety and traffic efficiency. For
example, detection of vehicles that violate traffic rules, parking enforcement,
etc
A Methodology for Studying VANET Performance with Practical Vehicle Distribution in Urban Environment
In a Vehicular Ad-hoc Network (VANET), the amount of interference from
neighboring nodes to a communication link is governed by the vehicle density
dynamics in vicinity and transmission probabilities of terminals. It is obvious
that vehicles are distributed non-homogeneously along a road segment due to
traffic controls and speed limits at different portions of the road. The common
assumption of homogeneous node distribution in the network in most of the
previous work in mobile ad-hoc networks thus appears to be inappropriate in
VANETs. In light of the inadequacy, we present in this paper an original
methodology to study the performance of VANETs with practical vehicle
distribution in urban environment. Specifically, we introduce the stochastic
traffic model to characterize the general vehicular traffic flow as well as the
randomness of individual vehicles, from which we can acquire the mean dynamics
and the probability distribution of vehicular density. As illustrative
examples, we demonstrate how the density knowledge from the stochastic traffic
model can be utilized to derive the throughput and progress performance of
three routing strategies in different channel access protocols. We confirm the
accuracy of the analytical results through extensive simulations. Our results
demonstrate the applicability of the proposed methodology on modeling protocol
performance, and shed insight into the performance analysis of other
transmission protocols and network configurations in vehicular networks.
Furthermore, we illustrate that the optimal transmission probability for
optimized network performance can be obtained as a function of the location
space from our results. Such information can be computed by road-side nodes and
then broadcasted to road users for optimized multi-hop packet transmission in
the communication network
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