2,484 research outputs found
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
Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group
Braided groups and braided matrices are novel algebraic structures living in
braided or quasitensor categories. As such they are a generalization of
super-groups and super-matrices to the case of braid statistics. Here we
construct braided group versions of the standard quantum groups . They
have the same FRT generators but a matrix braided-coproduct \und\Delta
L=L\und\tens L where , and are self-dual. As an application, the
degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices
; it is a braided-commutative bialgebra in a braided category. As a
second application, we show that the quantum double D(\usl) (also known as
the `quantum Lorentz group') is the semidirect product as an algebra of two
copies of \usl, and also a semidirect product as a coalgebra if we use braid
statistics. We find various results of this type for the doubles of general
quantum groups and their semi-classical limits as doubles of the Lie algebras
of Poisson Lie groups.Comment: 45 pages. Revised (= much expanded introduction
Coadditive differential complexes on quantum groups and quantum spaces
A regular way to define an additive coproduct (or ``coaddition'') on the
q-deformed differential complexes is proposed for quantum groups and quantum
spaces related to the Hecke-type R-matrices. Several examples of braided
coadditive differential bialgebras (Hopf algebras) are presented.Comment: 9 page
Quantum Deformations of Space-Time Symmetries with Mass-Like Deformation Parameter
The difficulties with the measurability of classical space-time distances are
considered. We outline the framework of quantum deformations of D=4 space-time
symmetries with dimensionfull deformation parameter, and present some recent
results.Comment: 4 pages, LaTeX, uses file stwol.sty, to be published in the
Proceedings of XXXII International Rochester Conference in High Energy
Physics (Warsaw, 24.07-31.07 1996
Force-induced rupture of a DNA duplex
The rupture of double-stranded DNA under stress is a key process in
biophysics and nanotechnology. In this article we consider the shear-induced
rupture of short DNA duplexes, a system that has been given new importance by
recently designed force sensors and nanotechnological devices. We argue that
rupture must be understood as an activated process, where the duplex state is
metastable and the strands will separate in a finite time that depends on the
duplex length and the force applied. Thus, the critical shearing force required
to rupture a duplex within a given experiment depends strongly on the time
scale of observation. We use simple models of DNA to demonstrate that this
approach naturally captures the experimentally observed dependence of the
critical force on duplex length for a given observation time. In particular,
the critical force is zero for the shortest duplexes, before rising sharply and
then plateauing in the long length limit. The prevailing approach, based on
identifying when the presence of each additional base pair within the duplex is
thermodynamically unfavorable rather than allowing for metastability, does not
predict a time-scale-dependent critical force and does not naturally
incorporate a critical force of zero for the shortest duplexes. Additionally,
motivated by a recently proposed force sensor, we investigate application of
stress to a duplex in a mixed mode that interpolates between shearing and
unzipping. As with pure shearing, the critical force depends on the time scale
of observation; at a fixed time scale and duplex length, the critical force
exhibits a sigmoidal dependence on the fraction of the duplex that is subject
to shearing.Comment: 10 pages, 6 figure
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
Statistics and UV-IR Mixing with Twisted Poincare Invariance
We elaborate on the role of quantum statistics in twisted Poincare invariant
theories. It is shown that, in order to have twisted Poincare group as the
symmetry of a quantum theory, statistics must be twisted. It is also confirmed
that the removal of UV-IR mixing (in the absence of gauge fields) in such
theories is a natural consequence.Comment: 13 pages, LaTeX; typos correcte
Reflection equations and q-Minkowski space algebras
We express the defining relations of the -deformed Minkowski space algebra
as well as that of the corresponding derivatives and differentials in the form
of reflection equations. This formulation encompasses the covariance properties
with respect the quantum Lorentz group action in a straightforward way.Comment: 10 page
Induced Representations of Quantum Kinematical Algebras and Quantum Mechanics
Unitary representations of kinematical symmetry groups of quantum systems are
fundamental in quantum theory. We propose in this paper its generalization to
quantum kinematical groups. Using the method, proposed by us in a recent paper
(olmo01), to induce representations of quantum bicrossproduct algebras we
construct the representations of the family of standard quantum inhomogeneous
algebras . This family contains the quantum
Euclidean, Galilei and Poincar\'e algebras, all of them in (1+1) dimensions. As
byproducts we obtain the actions of these quantum algebras on regular co-spaces
that are an algebraic generalization of the homogeneous spaces and --Casimir
equations which play the role of --Schr\"odinger equations.Comment: LaTeX 2e, 20 page
Quantum Groups and Noncommutative Geometry
Quantum groups emerged in the latter quarter of the 20th century as, on the
one hand, a deep and natural generalisation of symmetry groups for certain
integrable systems, and on the other as part of a generalisation of geometry
itself powerful enough to make sense in the quantum domain. Just as the last
century saw the birth of classical geometry, so the present century sees at its
end the birth of this quantum or noncommutative geometry, both as an elegant
mathematical reality and in the form of the first theoretical predictions for
Planck-scale physics via ongoing astronomical measurements. Noncommutativity of
spacetime, in particular, amounts to a postulated new force or physical effect
called cogravity.Comment: 72 pages, many figures; intended for wider theoretical physics
community (special millenium volume of JMP
- …