Intrinsic curvature of the 3-surfaces of homothety in spatially homothetic space-times

It is shown how the sign of the intrinsic scalar curvature of spatially homothetic space-times depends on their algebraic type, thus generalizing a corresponding result for the homogeneous case.     

Conformal groups and conformally equivalent isometry groups

It is shown that if ann dimensional Riemannian or pseudo-Riemannian manifold admits a proper conformal scalar, every (local) conformal group is conformally isometric, and that if it admits a proper conformal gradient every (local) conformal group is conformally homothetic. In the Riemannian case there is always a conformal scalar unless the metric is conformally Euclidean. In the case of a Lorentzian 4-manifold it is proved that the only metrics with no conformal scalars (and hence the only ones admitting a (local) conformal group not conformally isometric) are either conformal to the plane wave metric with parallel rays or conformally Minkowskian.     

D-set and groups

Kôpka and Chovanec have defined the concept of a D-poset, a partially ordered set with a partial operation with properties analogous to subtraction on the real line. In this paper we study similar structures, but we do not assume a partial order relation or the existence of distinguished elements 0, 1. We call each such structure a D-set and show that if a certain condition is satisfied, a D-set becomes the union of Abelian groups.     

Inhomogeneous quantum groups

Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsU _q (N), SO _q (N) and theq-deformed Lorentz group acting on affine quantum spaces are constructed. Special representations of the translation part are investigated.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.     

Inhomogeneous quantum groups

Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsU _q (N), SO _q (N) and theq-deformed Lorentz group acting on affine quantum spaces are constructed.     

Holonomy groups andW-symmetries

Irreducible sigma models, i.e. those for which the partition function does not factorise, are defined on Riemannian spaces with irreducible holonomy groups. These special geometries are characterised by the existence of covariantly constant forms which in turn give rise to symmetries of the supersymmetric sigma model actions. The Poisson bracket algebra of the corresponding currents is aW-algebra. Extended supersymmetries arise as special cases.     

Linear kinematical groups

We prove a theorem which states that in an (n+1)-dimensional space-time (n3) the only linear kinematical groups which are compatible with the isotropy of space are the Lorentz and Galilei groups. The special casesn=1 andn=2 are also briefly discussed.On leave of absence from Instituto di Fisica dell' Universitá, Milano, Italy. A. v. Humboldt fellow.     

The isometry groups of simply connected 3-dimensional unimodular Lie groups

For each simply connected 3-dimensional unimodular Lie group the left invariant Riemannian metrics up to automorphism are classified in Ha and Lee (2009) [2]. Using this classification, we determine on each such a Lie group the full group of isometries which depends on left invariant Riemannian metrics.     

A note on embedding non-Abelian finite flavor groups in continuous groups

A non-Abelian finite flavor group G⊂SO(3)$G\subset \mathit{SO}(3)$ can have double covering G^′⊂SU(2)$G^{\prime }\subset \mathit{SU}(2)$ such that G⊄G^′$G\not\subset G^{\prime }$. This situation is not contradictory, but quite natural, and we give explicit examples such as G=Dn$G=D_n$, G^′=Q_2n$G^{\prime }=Q_{2n}$ and G=T$G=T$, G^′=T^′$G^{\prime }=T^{\prime }$. This observation can be crucial in particle theory model building.     

From groups to bands

Boojums all the Way Through: Communicating Science in a Prosaic Age. By N. David Mermin. 1980, £30.00 (hbk), £10.95 (pbk), pp. xxi + 308. Cambridge University Press, ISBN (hbk) 0 521 38231 9 (pbk) 0 521 38880 5. Scope: Collected essays. Level: General reader.

From a Life of Physics. By H. A. Bethe, P. A. M. Dirac, W. Heisenberg, E. P. Wigner, O. Klein, L. D. Landau (edited by E. M. Lifshitz). 1989, £4.00, pp. ix + 92. World Scientific, ISBN 9971 50 937 7. Scope: Collected talks. Level: General reader.

Fractal Geometry. Mathematical Foundations and Applications. By Kenneth Falconer. 1990, £19.95, pp. xxii + 228. John Wiley, ISBN 0471 922870. Scope: Textbook. Level: Undergraduate.

Dynamical Systems and Fractals: Computer graphics experiments in Pascal. By Karl-Heinz Becker and Michael Döurfler. 1989, £30.00 (hbk), £10.95 (pbk), pp. xii + 398. Cambridge University Press, ISBN 0 52136025 0 (hbk), 0521 36910 X (pbk). Scope: Textbook. Level: Elementary.

Inverse Problems of Mathematical Physics. By V. B. Glasko. 1989, £20.50, pp. ix + 97. Adam Hilger, ISBN 088318571 7. Scope: Textbook. Level: Specialist.

Quantum Mechanics: Symmetries. By W. Greiner and B. Müller 1989, DM 78.00 (soft cover), pp. xvi + 368. Springer, ISBN 3 540 19201 8. Scope: Textbook. Level: Advanced undergraduate/postgraduate.

Old and New Forces of Nature. Edited by Antonino Zichichi. 1988, US$79.50, pp. viii + 382. Plenum Press, ISBN 0306 43235 8. Scope: Conference Proceedings. Level: Specialist.

Quantum Theory and Pictures of Reality: Foundations, Interpretations, and New Aspects. Edited by W. Schommers. 1989, DM 98.00, pp. xvii + 336. Springer, ISBN 3 540 50152 5. Scope: Multiauthor Reviews. Level: Specialist.

From Quarks to the Cosmos: Tools of Discovery. Leon M. Lederman and David N. Schramm. 1989, £16.95, pp. 241. W. H. Freeman &; Company, ISBN 0 7167 5052 X. Scope: Text. Level: General reader.

Babbles, Voids and Bumps in Time: The new cosmology. Edited by James Cornell. 1989, £12.95 (hbk), pp. xiii + 190. Cambridge University Press, ISBN 0 521 35297 5. Scope: General readership. Level: Undergraduate/VI-Form.

Cosmology and Elementary Particles. Edited by Daniel R. Altschuler and Jose F. Nieves. 1989, £38.00, pp. vii + 370. World Scientific, ISBN 9971 50 654 8. Scope: Multiauthor Reviews. Level: Postgraduate.

The Moon Observers Handbook. By Fred W. Price. 1989, £19.50, pp. xiii + 309. Cambridge University Press, ISBN 0 521 33500 0. Scope: Textbook. Level: General reader.

The Supernova Search Charts and Handbook. By G. D. Thompson and J. T. Bryan, Jr. 1990, £65.00 (hardback), pp. xx + 134. Cambridge University Press, ISBN 0 521 26721 8. Scope: Text. Level: General reader/Specialist.

Data in Astronomy. By Carlos Jaschek. 1989, £30.00, pp. 198. Cambridge University Press, ISBN 0 521 34094 2. Scope: Textbook. Level: Specialist.

Plasma Physics and Plasma Electronics. Edited by L. M. Kovrizhnykh. 1989, US$98.00, pp. x + 313. Nova Science, ISBN 0 941743 47 0. Scope: Textbook. Level: Specialist.

Transport Processes in Plasmas. Vol. 1. Classical Transport Theory. By R. Balescu. 1988, US $94.75, Dfl. 180.00, pp. xxxiii + 331. North-Holland, ISBN 0 444 87091 1. Scope: Monograph. Level: Postgraduate.

Transport Processes in Plasmas. Vol. 2. Neoclassical Transport. By R. Balescu. 1988, US $131.50, Dfl 250.00, pp. xxxiii + 333–802. North-Holland, ISBN 0 444 87092 X. Scope: Monograph. Level: Postgraduate.

The Supercontinuum Laser Source. Edited by Robert R. Alfano. 1989, DM 164.00, pp. xiv + 458. Springer, ISBN 3 540 96946 2. Scope: Multiauthor Reviews. Level: Specialist.

The Physics and Technology of Laser Resonators. Edited by D. R. Hall and P. E. Jackson. 1989, £25.00, pp. xiv + 253. Adam Hilger, ISBN 0 85274 1170. Scope: Multiauthor Review. Level: Postgraduate specialist.

High Gain, High Power Free Electron Laser: Physics and Application to TeV Particle Acceleration. Edited by R. Bonifacio, L. De Salvo Souva, C. Pellegrini. 1989, Dfl 185.00, pp. xv + 309. North-Holland, ISBN 0 444 87395 3. Scope: Conference Proceedings. Level: Postgraduate.

Nonlinear Fiber Optics. By Govind P. Agrawal. 1989, £39.95, pp. xii + 342. Academic Press, ISBN 0 12 045140 9. Scope: Textbook. Level: Postgraduate.

The Principles and Practice of Electron Microscopy. By Ian M. Watt. 1989, £20.00 (pbk), pp. viii + 304. Cambridge University Press, ISBN 0 521 38927 5. Scope: Textbook. Level: Postgraduate.

Auger Microprobe Analysis. By Ian F. Ferguson. 1989, £60.00, pp. x + 456. Adam Hilger, ISBN 0 85274 147 2. Scope: Textbook. Level: Postgraduate.

Gamma-Ray and Electron Spectroscopy in Nuclear Physics. By H. Ejiri and M. J. A. De Voigt. 1989, £75.00, pp. viii + 696. Clarendon Press, ISBN 0 19 851723 8. Scope: Textbook. Level: Postgraduate.

Extended Icosahedral Structures. Edited by Marko V. Jari? and Denis Gratias. 1989, £42.50, pp. x + 224. Academic Press, Inc. ISBN 0 12 0406039. Scope: Research Review. Level: Advanced.

Modern Vacuum Practice. By Nigel Harris. 1989, £24.95, pp. xiii + 315. McGraw Hill, ISBN 0 07 707099 2. Scope: Textbook. Level: Postgraduate.     

Causal groups of space-time

The present paper is concerned with a finite dimensional space, considered as a real ordered linear space, directed with respect to a partial ordering relation (causal relation) in which a given reflexion (called temporal inversion) is antiisotone and the positive cone is closed in the euclidean topology.A generalized Zeeman theorem [1] is obtained, which states that the causal group relative to the causal relation is a subgroup of the affine group ofM.     

Homothety groups in space-time

This paper investigates space-times which admit anr-dimensional Lie algebra of homothetic vector fields at least one of which is proer homothetic. The situation is resolved completely ifr 6 and some general results are given whenr 5. Some applications to the study of affine vector fields in space-times are also given and the maximum dimension of this latter algebra is computed and corrects an earlier error.     

Remarks on quantum groups

We give a Poisson-bracket realization of SL _q (2) in the phase space ². We then discuss the physical meaning of such a realization in terms of a modified (regularized) toy model, the nonregularized version of which is due to Klauder.Some general remarks and suggestions are also presented in this Letter.     

Duality and quantum groups

We show that the duality properties of Rational Conformal Field Theories follow from the defining relations and the representation theory of quantum groups. The fusion and braiding matrices are q-analogues of the 6j-symbols and the modular transformation matrices are obtained from the properties of the co-multiplication. We study in detail the Wess-Zumino-Witten models and the rational gaussian models as examples, but carry out the arguments in general. We point out the connections with the Chern-Simons approach. We give general arguments of why the general solution to the polynomial equations of Moore and Seiberg describing the duality properties of Rational Conformal Field Theories defines a Quantum Group acting on the space of conformal blocks. A direct connection between Rational Theories and knot invariants is also presented along the lines of Jones' original work.     

Space-time and holonomy groups

The possibilities for the (full) holonomy group of a (simply connected) vacuum space-time are calculated. The non-simply connected case is also discussed.     

Graphs and reflection groups

It is shown that graphs that generalize the ADE Dynkin diagrams and have appeared in various contexts of two-dimensional field theory may be regarded in a natural way as encoding the geometry of a root system. After recalling what are the conditions satisfied by these graphs, we define a bilinear form on a root system in terms of the adjacency matrices of these graphs and undertake the study of the group generated by the reflections in the hyperplanes orthogonal to these roots. Some non-integrally laced graphs are shown to be associated with subgroups of these reflection groups. The empirical relevance of these graphs in the classification of conformal field theories or in the construction of integrable lattice models is recalled, and the connections with recent developments in the context of supersymmetric theories and topological field theories are discussed. A. ClaudeDedicated to: A. Claude