48 research outputs found
Dirac Operator on the Quantum Sphere
We construct a Dirac operator on the quantum sphere which is
covariant under the action of . It reduces to Watamuras' Dirac
operator on the fuzzy sphere when . We argue that our Dirac operator
may be useful in constructing invariant field theories on
following the Connes-Lott approach to noncommutative geometry.Comment: 13 page
Induced representations of quantum kinematical algebras
We construct the induced representations of the null-plane quantum Poincar\'e
and quantum kappa Galilei algebras in (1+1) dimensions. The induction procedure
makes use of the concept of module and is based on the existence of a pair of
Hopf algebras with a nondegenerate pairing and dual bases.Comment: 8 pages,LaTeX2e, to be published in the Proceedings of XXIII
International Colloquium on Group-Theoretical Methods in Physics, Dubna
(Russia), 31.07--05.08, 200
Generalized Noiseless Quantum Codes utilizing Quantum Enveloping Algebras
A generalization of the results of Rasetti and Zanardi concerning avoiding
errors in quantum computers by using states preserved by evolution is
presented. The concept of dynamical symmetry is generalized from the level of
classical Lie algebras and groups to the level of dynamical symmetry based on
quantum Lie algebras and quantum groups (in the sense of Woronowicz). A natural
connection is proved between states preserved by representations of a quantum
group and states preserved by evolution with dynamical symmetry of the
appropriate universal enveloping algebra. Illustrative examples are discussed.Comment: 10 pages, LaTeX, 2 figures Postscrip
Quantum E(2) groups and Lie bialgebra structures
Lie bialgebra structures on are classified. For two Lie bialgebra
structures which are not coboundaries (i.e. which are not determined by a
classical -matrix) we solve the cocycle condition, find the Lie-Poisson
brackets and obtain quantum group relations. There is one to one correspondence
between Lie bialgebra structures on and possible quantum deformations of
and .Comment: 8 pages, plain TEX, harvmac, to appear in J. Phys.
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
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
Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group
We analyze the hamiltonian quantization of Chern-Simons theory associated to
the universal covering of the Lorentz group SO(3,1). The algebra of observables
is generated by finite dimensional spin networks drawn on a punctured
topological surface. Our main result is a construction of a unitary
representation of this algebra. For this purpose, we use the formalism of
combinatorial quantization of Chern-Simons theory, i.e we quantize the algebra
of polynomial functions on the space of flat SL(2,C)-connections on a
topological surface with punctures. This algebra admits a unitary
representation acting on an Hilbert space which consists in wave packets of
spin-networks associated to principal unitary representations of the quantum
Lorentz group. This representation is constructed using only Clebsch-Gordan
decomposition of a tensor product of a finite dimensional representation with a
principal unitary representation. The proof of unitarity of this representation
is non trivial and is a consequence of properties of intertwiners which are
studied in depth. We analyze the relationship between the insertion of a
puncture colored with a principal representation and the presence of a
world-line of a massive spinning particle in de Sitter space.Comment: 78 pages. Packages include
Twisted Classical Poincar\'{e} Algebras
We consider the twisting of Hopf structure for classical enveloping algebra
, where is the inhomogenous rotations algebra, with
explicite formulae given for Poincar\'{e} algebra
The comultiplications of twisted are obtained by conjugating
primitive classical coproducts by where
denotes any Abelian subalgebra of , and the universal
matrices for are triangular. As an example we show that
the quantum deformation of Poincar\'{e} algebra recently proposed by Chaichian
and Demiczev is a twisted classical Poincar\'{e} algebra. The interpretation of
twisted Poincar\'{e} algebra as describing relativistic symmetries with
clustered 2-particle states is proposed.Comment: \Large \bf 19 pages, Bonn University preprint, November 199
Phase spaces related to standard classical -matrices
Fundamental representations of real simple Poisson Lie groups are Poisson
actions with a suitable choice of the Poisson structure on the underlying
(real) vector space. We study these (mostly quadratic) Poisson structures and
corresponding phase spaces (symplectic groupoids).Comment: 20 pages, LaTeX, no figure
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