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 $U_h g$ of the Hopf algebra $Ug$. We propose a systematic method to construct all the corresponding deforming maps, together with the corresponding realizations of the action of $U_h g$. The method is then generalized and explicitly applied to the case that $U_h g$ is the quantum group $U_h sl(2)$. A preliminary study of the status of deforming maps at the representation level shows in particular that `deformed' Fock representations induced by a compact $U_h g$ 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): $\beta^*=\beta^{-1}$ 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