44 research outputs found
BRST Algebra Quantum Double and Quantization of the Proper Time Cotangent Bundle
The quantum double for the quantized BRST superalgebra is studied. The
corresponding R-matrix is explicitly constucted. The Hopf algebras of the
double form an analytical variety with coordinates described by the canonical
deformation parameters. This provides the possibility to construct the
nontrivial quantization of the proper time supergroup cotangent bundle. The
group-like classical limit for this quantization corresponds to the generic
super Lie bialgebra of the double.Comment: 11 pages, LaTe
Extended jordanian twists for Lie algebras
Jordanian quantizations of Lie algebras are studied using the factorizable
twists. For a restricted Borel subalgebras of the
explicit expressions are obtained for the twist element , universal
-matrix and the corresponding canonical element . It is
shown that the twisted Hopf algebra is
self dual. The cohomological properties of the involved Lie bialgebras are
studied to justify the existence of a contraction from the Dinfeld-Jimbo
quantization to the jordanian one. The construction of the twist is generalized
to a certain type of inhomogenious Lie algebras.Comment: 28 pages, LaTe
Quantum Jordanian twist
The quantum deformation of the Jordanian twist F_qJ for the standard quantum
Borel algebra U_q(B) is constructed. It gives the family U_qJ(B) of quantum
algebras depending on parameters x and h. In a generic point these algebras
represent the hybrid (standard-nonstandard) quantization. The quantum Jordanian
twist can be applied to the standard quantization of any Kac-Moody algebra. The
corresponding classical r-matrix is a linear combination of the Drinfeld- Jimbo
and the Jordanian ones. The obtained two-parametric families of Hopf algebras
are smooth and for the limit values of the parameters the standard and
nonstandard quantizations are recovered. The twisting element F_qJ also has the
correlated limits, in particular when q tends to unity it acquires the
canonical form of the Jordanian twist. To illustrate the properties of the
quantum Jordanian twist we construct the hybrid quantizations for U(sl(2)) and
for the corresponding affine algebra U(hat(sl(2))). The universal quantum
R-matrix and its defining representation are presented.Comment: 12 pages, Late
Three dimensional quantum algebras: a Cartan-like point of view
A perturbative quantization procedure for Lie bialgebras is introduced and
used to classify all three dimensional complex quantum algebras compatible with
a given coproduct. The role of elements of the quantum universal enveloping
algebra that, analogously to generators in Lie algebras, have a distinguished
type of coproduct is discussed, and the relevance of a symmetrical basis in the
universal enveloping algebra stressed. New quantizations of three dimensional
solvable algebras, relevant for possible physical applications for their
simplicity, are obtained and all already known related results recovered. Our
results give a quantization of all existing three dimensional Lie algebras and
reproduce, in the classical limit, the most relevant sector of the complete
classification for real three dimensional Lie bialgebra structures given by X.
Gomez in J. Math. Phys. Vol. 41. (2000) 4939.Comment: LaTeX, 15 page
Classical Dynamical Systems from q-algebras:"cluster" variables and explicit solutions
A general procedure to get the explicit solution of the equations of motion
for N-body classical Hamiltonian systems equipped with coalgebra symmetry is
introduced by defining a set of appropriate collective variables which are
based on the iterations of the coproduct map on the generators of the algebra.
In this way several examples of N-body dynamical systems obtained from
q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2)
Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of
Ruijsenaars type arising from the same (non co-boundary) q-deformation of the
(1+1) Poincare' algebra. Also, a unified interpretation of all these systems as
different Poisson-Lie dynamics on the same three dimensional solvable Lie group
is given.Comment: 19 Latex pages, No figure
Semi-classical twists for sl(3) and sl(4) boundary r-matrices of Cremmer-Gervais type
We obtain explicit formulas for the semi-classical twists deforming the
coalgebraic structure of U(sl(3)) and U(sl(4)). In rank 2 and 3 the
corresponding universal R-matrices quantize the boundary r-matrices of
Cremmer-Gervais type defining Lie Frobenius structures on the maximal parabolic
subalgebras in sl(n)
Peripheric Extended Twists
The properties of the set L of extended jordanian twists are studied. It is
shown that the boundaries of L contain twists whose characteristics differ
considerably from those of internal points. The extension multipliers of these
"peripheric" twists are factorizable. This leads to simplifications in the
twisted algebra relations and helps to find the explicit form for coproducts.
The peripheric twisted algebra U(sl(4)) is obtained to illustrate the
construction. It is shown that the corresponding deformation U_{P}(sl(4))
cannot be connected with the Drinfeld--Jimbo one by a smooth limit procedure.
All the carrier algebras for the extended and the peripheric extended twists
are proved to be Frobenius.Comment: 16 pages, LaTeX 209. Some misprints have been corrected and new
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A damage model based on failure threshold weakening
A variety of studies have modeled the physics of material deformation and
damage as examples of generalized phase transitions, involving either critical
phenomena or spinodal nucleation. Here we study a model for frictional sliding
with long range interactions and recurrent damage that is parameterized by a
process of damage and partial healing during sliding. We introduce a failure
threshold weakening parameter into the cellular-automaton slider-block model
which allows blocks to fail at a reduced failure threshold for all subsequent
failures during an event. We show that a critical point is reached beyond which
the probability of a system-wide event scales with this weakening parameter. We
provide a mapping to the percolation transition, and show that the values of
the scaling exponents approach the values for mean-field percolation (spinodal
nucleation) as lattice size is increased for fixed . We also examine the
effect of the weakening parameter on the frequency-magnitude scaling
relationship and the ergodic behavior of the model
Duality for Exotic Bialgebras
In the classification of Hietarinta, three triangular
-matrices lead, via the FRT formalism, to matrix bialgebras which are not
deformations of the trivial one. In this paper, we find the bialgebras which
are in duality with these three exotic matrix bialgebras. We note that the
duality of FRT is not sufficient for the construction of the bialgebras
in duality. We find also the quantum planes corresponding to these bialgebras
both by the Wess-Zumino R-matrix method and by Manin's method.Comment: 25 pages, LaTeX2e, using packages: cite, amsfonts, amsmath, subeq
Multiparametric quantum gl(2): Lie bialgebras, quantum R-matrices and non-relativistic limits
Multiparametric quantum deformations of are studied through a
complete classification of Lie bialgebra structures. From them, the
non-relativistic limit leading to harmonic oscillator Lie bialgebras is
implemented by means of a contraction procedure. New quantum deformations of
together with their associated quantum -matrices are obtained and
other known quantizations are recovered and classified. Several connections
with integrable models are outlined.Comment: 21 pages, LaTeX. To appear in J. Phys. A. New contents adde