18 research outputs found
Deforming Maps for Lie Group Covariant Creation and Annihilation Operators
Any deformation of a Weyl or Clifford algebra A can be realized through a
`deforming map', i.e. a formal change of generators in A. This is true in
particular if A is covariant under a Lie algebra g and its deformation is
induced by some triangular deformation of the Hopf algebra . We
propose a systematic method to construct all the corresponding deforming maps,
together with the corresponding realizations of the action of . The
method is then generalized and explicitly applied to the case that is
the quantum group . A preliminary study of the status of deforming
maps at the representation level shows in particular that `deformed' Fock
representations induced by a compact can be interpreted as standard
`undeformed' Fock representations describing particles with ordinary Bose or
Fermi statistics.Comment: Latex file, 26 pages, no figures. Extended changes. Final Version to
appear in J. Math. Phy
q-Deforming Maps for Lie Group Covariant Heisenberg Algebras
We briefly summarize our systematic construction procedure of q-deforming
maps for Lie group covariant Weyl or Clifford algebras.Comment: latex file, 4 pages. Contribution to the proceedings of the 5th
Wigner Symposium. Slight modification
On Two Theorems About Symplectic Reflection Algebras
We give a new proof and an improvement of two Theorems of J. Alev, M.A.
Farinati, T. Lambre and A.L. Solotar : the first one about Hochschild
cohomology spaces of some twisted bimodules of the Weyl algebra W and the
second one about Hochschild cohomology spaces of the smash product G * W (G a
finite subgroup of SP(2n)), and as an application, we then give a new proof of
a Theorem of P. Etingof and V. Ginzburg, which shows that the Symplectic
Reflection Algebras are deformations of G * W (and, in fact, all possible
ones).Comment: corrected typo
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
On second quantization on noncommutative spaces with twisted symmetries
By application of the general twist-induced star-deformation procedure we
translate second quantization of a system of bosons/fermions on a symmetric
spacetime in a non-commutative language. The procedure deforms in a coordinated
way the spacetime algebra and its symmetries, the wave-mechanical description
of a system of n bosons/fermions, the algebra of creation and annihilation
operators and also the commutation relations of the latter with functions of
spacetime; our key requirement is the mode-decomposition independence of the
quantum field. In a conservative view, the use of noncommutative coordinates
can be seen just as a way to better express non-local interactions of a special
kind. In a non-conservative one, we obtain a covariant framework for QFT on the
corresponding noncommutative spacetime consistent with quantum mechanical
axioms and Bose-Fermi statistics. One distinguishing feature is that the field
commutation relations remain of the type "field (anti)commutator=a
distribution". We illustrate the results by choosing as examples interacting
non-relativistic and free relativistic QFT on Moyal space(time)s.Comment: Latex file, 45 pages. I have corrected a small typo present in 3
points of the previous version and in the version published also in JPA
(which had occurred via late careless serial replacements, with no
consequences on the results of the calculations): has
been corrected into $\beta^*=S(\beta^{-1})
Geodesic rewriting systems and pregroups
In this paper we study rewriting systems for groups and monoids, focusing on
situations where finite convergent systems may be difficult to find or do not
exist. We consider systems which have no length increasing rules and are
confluent and then systems in which the length reducing rules lead to
geodesics. Combining these properties we arrive at our main object of study
which we call geodesically perfect rewriting systems. We show that these are
well-behaved and convenient to use, and give several examples of classes of
groups for which they can be constructed from natural presentations. We
describe a Knuth-Bendix completion process to construct such systems, show how
they may be found with the help of Stallings' pregroups and conversely may be
used to construct such pregroups.Comment: 44 pages, to appear in "Combinatorial and Geometric Group Theory,
Dortmund and Carleton Conferences". Series: Trends in Mathematics.
Bogopolski, O.; Bumagin, I.; Kharlampovich, O.; Ventura, E. (Eds.) 2009,
Approx. 350 p., Hardcover. ISBN: 978-3-7643-9910-8 Birkhause