Star products for SL(2)

The deformation program (the use of star products in harmonic analysis) leads to the definition of an adapted Fourier transform, unitary transformation between spaces of square integrable functions of the group G and on the dual of its Lie algebra, describing the unitary dual of G and its Plancherel transform. This Letter is an application of this program to the universal covering group of SL(2).     

Solitons and SL(2, R)

The known relationship between non-linear partial differential equations which have soliton solutions, and SL (2, R), is developed to the point where it provides a framework for discussing Bäcklund transformations, and equations for the inverse scattering method.     

SL(4, R)-invariant chiral fields

Using a method developed before a set of exact solutions of the chiral equations , wheregSL(4,R) are presented.Work supported in part by CONACYT, México.     

SL(4,R) classification for hadrons

It is suggested that the complete spectrum of baryon and meson resonances for each flavour is given by spinor and tensor infinite-component systems based on unitary irreducible representations of $\text{SL}$(4,R). This is a shell-model-like dynamical geometrical symmetry, presumably resulting from the quark structure and QCD, with possible connections to gravity. The fit to observations is excellent.     

SL(2,C) gauge theory of gravitation: Conservation laws

Conservation laws in the SL(2,C) gauge theory of gravitation are reviewed and their relation to the ordinary conservation laws in general relativity theory is discussed. The vector currents that were proposed by different authors along the lines of the Yang-Mills conserved vector current are discussed and their interrelation is given. Likewise Lagrangian densitiies, from which one obtains the SL(2,C) gauge theory gravitational field equations, are discussed and related to the conservation laws through Noether's theorem.     

SL(2C) invariance of the gravitational field

The gravitational field equations of general relativity theory are cast into a Yang-Mills-type theory by use of the group SL(2,C). The spin coefficients take the rôle of the Yang-Mills-like potentials, whereas the Riemann tensor takes the rôle of the fields. Comparison of this formalism with that of Utiyama and Kibble who related invariance under the Lorentz and the Poincaré groups to the existence of the gravitational field, is discussedBased on a lecture given at the International Conference on Relativity and Gravitation, Copenhagen, July 1971.     

SU(6) × SL(3, R) color quark model

A nonrelativistic color quark model based on the spectrum generating group SU(6)×SL(3, R) is presented. The model unifies SU(6) and the Regge classification with linear L vs m² mass spectrum. Baryons are predicted to belong to the ($\text{56}\text{, L}{}^{\mathrm{P}}\text{= (}\text{even}{}^{\mathrm{+}}\text{)}\text{and}\text{(}\text{70}\text{, L}{}^{\mathrm{P}}\text{= (}\text{odd}{}^{\mathrm{?}}\text{)}$ multiplets of the SU (6) × O(3) group.     

Higher order differential calculus on SL q(N)

Let be a bicovariant first order differential calculus on a Hopf algebra . There are three possibilities to construct a differential N ₀-graded Hopf algebra which contains as its first order part. In all cases is a quotient = /J of the tensor algebra by some suitable ideal. We distinguish three possible choices _u J, _s J, and _W J, where the first one generates the universal differential calculus (over ) and the last one is Woronowicz' external algebra. Let q be a transcendental complex number and let be one of the N ²-dimensional bicovariant first order differential calculi on the quantum group SL _q(N). Then for N 3 the three ideals coincide. For Woronowicz' external algebra we calculate the dimensions of the spaces of left-invariant and bi-invariant k-forms. In this case each bi-invariant form is closed. In case of 4D _± calculi on SL _q(2) the universal calculus is strictly larger than the other two calculi. In particular, the bi-invariant 1-form is not closed.     

Nuclear rotational bands and SL(3, R) symmetry

Nuclear rotational motion is studied under the assumption that the transition E2 operators generate the algebraic symmetry SL(3, R), as first suggested by Gell-Mann, Ne'eman and Dothan. Besides the K = 0⁺ and K = 0⁻ bands, more general multiband representations are derived. This model predicts ratios of interband transitions in rotational nuclei having no adjustable parameters. These predictions are compared to all available data and show that the model is a semi-quantitative improvement on the rigid-rotator model.     

Bicovariant differential calculi on SL q(N) and Sp q(N)

For transcendental values of the deformation parameter q all bicovariant first order differential calculi on the Hopf algebras are classified.     

Unitarity of SL(2)-conformal blocks in genus zero

It has been conjectured for quite some time that a bundle of conformal blocks carries a unitary structure that is (projectively) flat for the Hitchin connection. This was recently established by T.R. Ramadas in the simplest nontrivial case, namely where the genus is zero and the group is SL(2). In this paper we present a shorter and more direct version of his proof. We also identify the conformal block space with the bidegree (N$N$, 0)-part of an eigenspace of a finite group acting on a Hodge structure of weight N$N$.     

SL(2,C)-gauge theory andN=1 supergravity

It is shown that as Riemannian space may be taken to give rise to a Poincaré gauge theory of gravitation, the superspace where the coordinates are given by (X, ), being a spinorial variable gives rise to anSL(2, C)-gauge theory and corresponds toN= 1 supergravity. It leads to a conserved current and the conserved quantity here corresponds to isospin, where the latter is taken to be generated from conformal reflection. Thus, supergravity plays a predominant role in the microlocal space-time.     

On Separation of Variables for Homogeneous SL(r) Gaudin Systems

By means of a recently introduced bihamiltonian structure for the homogeneous Gaudin models, we find a new set of Separation Coordinates for the sl(r) case.