5,365 research outputs found
Quantum-sl(2) action on a divided-power quantum plane at even roots of unity
We describe a nonstandard version of the quantum plane, the one in the basis
of divided powers at an even root of unity . It can be regarded
as an extension of the "nearly commutative" algebra with by nilpotents. For this quantum plane, we construct a Wess--Zumino-type de
Rham complex and find its decomposition into representations of the
-dimensional quantum group and its Lusztig extension; the
quantum group action is also defined on the algebra of quantum differential
operators on the quantum plane.Comment: 18 pages, amsart++, xy, times. V2: a reference and related comments
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Quantum Deformations of Conformal Algebras Introducing Fundamental Mass Parameters
We consider new class of classical r-matrices for D=3 and D=4 conformal Lie
algebras. These r-matrices do satisfy the classical Yang-Baxter equation and as
two-tensors belong to the tensor product of Borel subalgebra. In such a way we
generalize the lowest order of known nonstandard quantum deformation of sl(2)
to the Lie algebras sp(4) = so(5) and sl(4) = so(6). As an exercise we
interpret nonstandard deformation of sl(2) as describing quantum D=1 conformal
algebra with fundamental mass parameter. Further we describe the D=3 and D=4
conformal bialgebras with deformation parameters equal to the inverse of
fundamental masses. It appears that for D=4 the deformation of the Poincare
algebra sector coincides with "null plane" quantum Poincare algebra.Comment: LaTeX source, 12 page
Two-parameter nonstandard deformation of 2x2 matrices
We introduce a two-parameter deformation of 2x2 matrices without imposing any
condition on the matrices and give the universal R-matrix of the nonstandard
quantum group which satisfies the quantum Yang-Baxter relation. Although in the
standard two-parameter deformation the quantum determinant is not central, in
the nonstandard case it is central. We note that the quantum group thus
obtained is related to the quantum supergroup by a
transformation.Comment: 10 page
Quantum planes and quantum cylinders from Poisson homogeneous spaces
Quantum planes and a new quantum cylinder are obtained as quantization of
Poisson homogeneous spaces of two different Poisson structures on classical
Euclidean group E(2).Comment: 13 pages, plain Tex, no figure
RTT relations, a modified braid equation and noncommutative planes
With the known group relations for the elements of a quantum
matrix as input a general solution of the relations is sought without
imposing the Yang - Baxter constraint for or the braid equation for
. For three biparametric deformatios, and , the standard,the nonstandard and the
hybrid one respectively, or is found to depend, apart from the
two parameters defining the deformation in question, on an extra free parameter
,such that only for two values of , given explicitly for each case, one
has the braid equation. Arbitray corresponds to a class (conserving the
group relations independent of ) of the MQYBE or modified quantum YB
equations studied by Gerstenhaber, Giaquinto and Schak. Various properties of
the triparametric , and are
studied. In the larger space of the modified braid equation (MBE) even
can satisfy outside braid equation (BE)
subspace. A generalized, - dependent, Hecke condition is satisfied by each
3-parameter . The role of in noncommutative geometries of the
, and deformed planes is studied. K is found to
introduce a "soft symmetry breaking", preserving most interesting properties
and leading to new interesting ones. Further aspects to be explored are
indicated.Comment: Latex, 17 pages, minor change
Resonances for a Hydrogenic System or a Harmonic Oscillator Strongly Coupled to a Field
We calculate resonances which are formed by a particle in a potential which
is either Coulombian or quadratic when the particle is strongly coupled to a
massless boson, taking only two energy levels into consideration. From these
calculations we derive how the moving away of the particle from its attraction
center goes together with the energy lowering of hybrid states that this
particle forms with the field. We study the width of these states and we show
that stable states may also appear in the coupling.Comment: 17 pages, 6 figure
Quantization of Nonstandard Hamiltonian Systems
The quantization of classical theories that admit more than one Hamiltonian
description is considered. This is done from a geometrical viewpoint, both at
the quantization level (geometric quantization) and at the level of the
dynamics of the quantum theory. A spin-1/2 system is taken as an example in
which all the steps can be completed. It is shown that the geometry of the
quantum theory imposes restrictions on the physically allowed nonstandard
quantum theories.Comment: Revtex file, 23 pages, no figure
On a "New" Deformation of GL(2)
We refute a recent claim in the literature of a "new" quantum deformation of
GL(2).Comment: 4 pages, LATE
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