Twisting 2-cocycles for the construction of new non-standard quantum groups

We introduce a new class of 2-cocycles defined explicitly on the generators of certain multiparameter standard quantum groups. These allow us, through the process of twisting the familiar standard quantum groups, to generate new as well as previously known examples of non-standard quantum groups. In particular we are able to construct generalisations of both the Cremmer-Gervais deformation of SL(3) and the so called esoteric quantum groups of Fronsdal and Galindo in an explicit and straightforward manner.Comment: 21 pages, AMSLaTeX, expanded introduction and a few other minor corrections, to appear in JM

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

Translational-invariant noncommutative gauge theory

A generalized translational invariant noncommutative field theory is analyzed in detail, and a complete description of translational invariant noncommutative structures is worked out. The relevant gauge theory is described, and the planar and nonplanar axial anomalies are obtained.Comment: V1: 23 pages, 4 figures; V2: Section I. improved, References added. Version accepted for publication in PR

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

Examples of q-regularization

An Introduction to Hopf algebras as a tool for the regularization of relavent quantities in quantum field theory is given. We deform algebraic spaces by introducing q as a regulator of a non-commutative and non-cocommutative Hopf algebra. Relevant quantities are finite provided q\neq 1 and diverge in the limit q\rightarrow 1. We discuss q-regularization on different q-deformed spaces for \lambda\phi^4 theory as example to illustrate the idea.Comment: 17 pages, LaTex, to be published in IJTP 1995.1

On the Clebsch-Gordan coefficients for the two-parameter quantum algebra SU(2)p,qSU(2)_{p,q}

We show that the Clebsch - Gordan coefficients for the $SU(2)_{p,q}$ - algebra depend on a single parameter Q = $\sqrt{pq}$ ,contrary to the explicit calculation of Smirnov and Wehrhahn [J.Phys.A 25 (1992),5563].Comment: 5 page

Gauge invariant formulation of N=2N=2 Toda and KdV systems in extended superspace

We give a gauge invariant formulation of $N=2$ supersymmetric abelian Toda field equations in \n2 superspace. Superconformal invariance is studied. The conserved currents are shown to be associated with Drinfeld-Sokolov type gauges. The extension to non-abelian \n2 Toda equations is discussed. Very similar methods are then applied to a matrix formulation in \n2 superspace of one of the \n2 KdV hierarchies.Comment: 21 page

Quantum function algebras as quantum enveloping algebras

Inspired by a result in [Ga], we locate two $k[q,q^{-1}]$-integer forms of $F_q[SL(n+1)]$, along with a presentation by generators and relations, and prove that for $q=1$ they specialize to $U({\mathfrak{h}})$, where ${\mathfrak{h}}$ is the Lie bialgebra of the Poisson Lie group $H$ dual of $SL(n+1)$; moreover, we explain the relation with [loc. cit.]. In sight of this, we prove two PBW-like theorems for $F_q[SL(n+1)]$, both related to the classical PBW theorem for $U({\mathfrak{h}})$.Comment: 27 pages, AMS-TeX C, Version 3.0 - Author's file of the final version, as it appears in the journal printed version, BUT for a formula in Subsec. 3.5 and one in Subsec. 5.2 - six lines after (5.1) - that in this very pre(post)print have been correcte

Uniformizing the Stacks of Abelian Sheaves

Elliptic sheaves (which are related to Drinfeld modules) were introduced by Drinfeld and further studied by Laumon--Rapoport--Stuhler and others. They can be viewed as function field analogues of elliptic curves and hence are objects "of dimension 1". Their higher dimensional generalisations are called abelian sheaves. In the analogy between function fields and number fields, abelian sheaves are counterparts of abelian varieties. In this article we study the moduli spaces of abelian sheaves and prove that they are algebraic stacks. We further transfer results of Cerednik--Drinfeld and Rapoport--Zink on the uniformization of Shimura varieties to the setting of abelian sheaves. Actually the analogy of the Cerednik--Drinfeld uniformization is nothing but the uniformization of the moduli schemes of Drinfeld modules by the Drinfeld upper half space. Our results generalise this uniformization. The proof closely follows the ideas of Rapoport--Zink. In particular, analogies of $p$-divisible groups play an important role. As a crucial intermediate step we prove that in a family of abelian sheaves with good reduction at infinity, the set of points where the abelian sheaf is uniformizable in the sense of Anderson, is formally closed.Comment: Final version, appears in "Number Fields and Function Fields - Two Parallel Worlds", Papers from the 4th Conference held on Texel Island, April 2004, edited by G. van der Geer, B. Moonen, R. Schoo

Twisted Rindler space-times

The (linearized) noncommutative Rindler space-times associated with canonical, Lie-algebraic and quadratic twist-deformed Minkowski spaces are provided. The corresponding deformed Hawking spectra detected by Rindler observers are derived as well.Comment: 13 pages, no figures, keywords: quantum space-times, Hawking radiatio