By Denis Bernard (auth.), J. Fröhlich, G. ’t Hooft, A. Jaffe, G. Mack, P. K. Mitter, R. Stora (eds.)

Soon after the invention of quantum mechanics, workforce theoretical tools have been used commonly in an effort to make the most rotational symmetry and classify atomic spectra. And till lately it used to be notion that symmetries in quantum mechanics might be teams. however it isn't so. There are extra common algebras, outfitted with appropriate constitution, which admit a wonderfully traditional interpretation as a symmetry of a quantum mechanical method. as a minimum, a "trivial illustration" of the algebra is outlined, and a tensor fabricated from representations. yet against this with teams, this tensor product has to be neither commutative nor associative. Quantum teams are targeted circumstances, during which associativity is preserved. The exploitation of such "Quantum Symmetries" used to be a vital topic on the advert vanced examine Institute. Introductory lectures have been awarded to familiarize the members with the al gebras that can look as symmetries and with their homes. a few versions of neighborhood box theories have been mentioned intimately that have a few such symmetries, in par ticular conformal box theories and their perturbations. Lattice versions offer many examples of quantum theories with quantum symmetries. They have been additionally lined on the university. eventually, the symmetries that are the reason for the solubility of inte grable types also are quantum symmetries of this sort. a few such versions and their nonlocal conserved currents have been discussed.

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**Example text**

On the algebra of smooth functions on a compact Riemannian spin manifold, the Dirac operator determines a K cycle which is (d,oo) summable where d =dim X. If X is even-dimensional then the K cycle is even, with r being the chirality operator. We shall call this the Dirac K cycle. '1. is not commutative, or when D is no longer the Dirac operator. '1. is the * algebra of matrix-valued functions on a Riemannian manifold, just in order to have in mind that the usual notion of exterior product does not make sense in the latter case.

Plenum Press, New York, 1992 37 Fig. 2 This corresponds to a base-point choice. A state of this "pointed figure" is now by definition a marker "--t" placed in each remaining region, pointing to one of its vertexes - but with the proviso that each vertex receives only one pointer. Thus in our situation there are three possible states: S2 Fig. 3 Next associate a weight with each state as follows. The overpasses can be rotated to fit one of two types: ~ I Fig. 4 and one marks these alternatives according to the principle: Fig.

Moreover, as we will soon show, the comultiplications are those in Y(9). The three first relations are easily proved. The last relation is more interesting and can be proved in geometrical way. It consists in imposing a Lorentz boost 'R 21r of angle (i21l') to the non-local currents J(1):(x,t). It is a rotation of (21l') in the Euclidian plane. Because the currents J{l):(x, t) are non-local this transformation 27 does not act trivially on them: the string ex winds arOlmd the point x. 21) Integrating the time-component of eq.