1,048 research outputs found
Braidings of Tensor Spaces
Let be a braided vector space, that is, a vector space together with a
solution of the Yang--Baxter equation.
Denote . We associate to a solution
of the Yang--Baxter equation on
the tensor space . The correspondence is functorial with respect to .Comment: 10 pages, no figure
Bicrossed products for finite groups
We investigate one question regarding bicrossed products of finite groups
which we believe has the potential of being approachable for other classes of
algebraic objects (algebras, Hopf algebras). The problem is to classify the
groups that can be written as bicrossed products between groups of fixed
isomorphism types. The groups obtained as bicrossed products of two finite
cyclic groups, one being of prime order, are described.Comment: Final version: to appear in Algebras and Representation Theor
On the deformability of Heisenberg algebras
Based on the vanishing of the second Hochschild cohomology group of the
enveloping algebra of the Heisenberg algebra it is shown that differential
algebras coming from quantum groups do not provide a non-trivial deformation of
quantum mechanics. For the case of a q-oscillator there exists a deforming map
to the classical algebra. It is shown that the differential calculus on quantum
planes with involution, i.e. if one works in position-momentum realization, can
be mapped on a q-difference calculus on a commutative real space. Although this
calculus leads to an interesting discretization it is proved that it can be
realized by generators of the undeformed algebra and does not posess a proper
group of global transformations.Comment: 16 pages, latex, no figure
Equivalence of -bosons using the exponential phase operator
Various forms of the -boson are explained and their hidden symmetry
revealed by transformations using the exponential phase operator. Both the
one-component and the multicomponent -bosons are discussed. As a byproduct,
we obtain a new boson algebra having a shifted vacuum structure and define a
global operatal gauge transformation.Comment: 18 pages, LaTex(run twice), To appear in J. PHys.
Perturbative Symmetries on Noncommutative Spaces
Perturbative deformations of symmetry structures on noncommutative spaces are
studied in view of noncommutative quantum field theories. The rigidity of
enveloping algebras of semi-simple Lie algebras with respect to formal
deformations is reviewed in the context of star products. It is shown that
rigidity of symmetry algebras extends to rigidity of the action of the symmetry
on the space. This implies that the noncommutative spaces considered can be
realized as star products by particular ordering prescriptions which are
compatible with the symmetry. These symmetry preserving ordering prescriptions
are calculated for the quantum plane and four-dimensional quantum Euclidean
space. Using these ordering prescriptions greatly facilitates the construction
of invariant Lagrangians for quantum field theory on noncommutative spaces with
a deformed symmetry.Comment: 16 pages; LaTe
Factorizing twists and R-matrices for representations of the quantum affine algebra U_q(\hat sl_2)
We calculate factorizing twists in evaluation representations of the quantum
affine algebra U_q(\hat sl_2). From the factorizing twists we derive a
representation independent expression of the R-matrices of U_q(\hat sl_2).
Comparing with the corresponding quantities for the Yangian Y(sl_2), it is
shown that the U_q(\hat sl_2) results can be obtained by `replacing numbers by
q-numbers'. Conversely, the limit q -> 1 exists in representations of U_q(\hat
sl_2) and both the factorizing twists and the R-matrices of the Yangian Y(sl_2)
are recovered in this limit.Comment: 19 pages, LaTe
Quantum spin coverings and statistics
SL_q(2) at odd roots of unity q^l =1 is studied as a quantum cover of the
complex rotation group SO(3,C), in terms of the associated Hopf algebras of
(quantum) polynomial functions. We work out the irreducible corepresentations,
the decomposition of their tensor products and a coquasitriangular structure,
with the associated braiding (or statistics). As an example, the case l=3 is
discussed in detail.Comment: 15 page
Kitaev's quantum double model from a local quantum physics point of view
A prominent example of a topologically ordered system is Kitaev's quantum
double model for finite groups (which in particular
includes , the toric code). We will look at these models from
the point of view of local quantum physics. In particular, we will review how
in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the
different superselection sectors of the model. In this way one finds that the
charges are in one-to-one correspondence with the representations of
, and that they are in fact anyons. Interchanging two of such
anyons gives a non-trivial phase, not just a possible sign change. The case of
non-abelian groups is more complicated. We outline how one could use
amplimorphisms, that is, morphisms to study the superselection
structure in that case. Finally, we give a brief overview of applications of
topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J.
Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015).
Mainly revie
Quantum Liouville theory and BTZ black hole entropy
In this paper I give an explicit conformal field theory description of
(2+1)-dimensional BTZ black hole entropy. In the boundary Liouville field
theory I investigate the reducible Verma modules in the elliptic sector, which
correspond to certain irreducible representations of the quantum algebra
U_q(sl_2) \odot U_{\hat{q}}(sl_2). I show that there are states that decouple
from these reducible Verma modules in a similar fashion to the decoupling of
null states in minimal models. Because ofthe nonstandard form of the Ward
identity for the two-point correlation functions in quantum Liouville field
theory, these decoupling states have positive-definite norms. The explicit
counting from these states gives the desired Bekenstein-Hawking entropy in the
semi-classical limit when q is a root of unity of odd order.Comment: LaTeX, 33 pages, 4 eps figure
Non-Abelian Anyons and Topological Quantum Computation
Topological quantum computation has recently emerged as one of the most
exciting approaches to constructing a fault-tolerant quantum computer. The
proposal relies on the existence of topological states of matter whose
quasiparticle excitations are neither bosons nor fermions, but are particles
known as {\it Non-Abelian anyons}, meaning that they obey {\it non-Abelian
braiding statistics}. Quantum information is stored in states with multiple
quasiparticles, which have a topological degeneracy. The unitary gate
operations which are necessary for quantum computation are carried out by
braiding quasiparticles, and then measuring the multi-quasiparticle states. The
fault-tolerance of a topological quantum computer arises from the non-local
encoding of the states of the quasiparticles, which makes them immune to errors
caused by local perturbations. To date, the only such topological states
thought to have been found in nature are fractional quantum Hall states, most
prominently the \nu=5/2 state, although several other prospective candidates
have been proposed in systems as disparate as ultra-cold atoms in optical
lattices and thin film superconductors. In this review article, we describe
current research in this field, focusing on the general theoretical concepts of
non-Abelian statistics as it relates to topological quantum computation, on
understanding non-Abelian quantum Hall states, on proposed experiments to
detect non-Abelian anyons, and on proposed architectures for a topological
quantum computer. We address both the mathematical underpinnings of topological
quantum computation and the physics of the subject using the \nu=5/2 fractional
quantum Hall state as the archetype of a non-Abelian topological state enabling
fault-tolerant quantum computation.Comment: Final Accepted form for RM
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