By L. D. Faddeev (auth.), L. A. Ibort, M. A. Rodríguez (eds.)

In some ways the decade has witnessed a surge of curiosity within the interaction among theoretical physics and a few conventional components of natural arithmetic. This e-book comprises the lectures introduced on the NATO-ASI summer season university on `Recent difficulties in Mathematical Physics' held at Salamanca, Spain (1992), supplying a pedagogical and up to date method of a number of the difficulties which have been on the center of those occasions. between them, we must always point out the hot mathematical constructions regarding integrability and quantum box theories, reminiscent of quantum teams, conformal box theories, integrable statistical versions, and topological quantum box theories, which are mentioned at size by way of the various top specialists at the components in different of the lectures inside the e-book. except those, conventional and new difficulties in quantum gravity are reviewed. different contributions to the varsity incorporated within the e-book diversity from symmetries in partial differential equations to geometrical levels in quantum physics. The publication is addressed to researchers within the fields lined, PhD scholars and any scientist drawn to acquiring an up to date view of the subjects.

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Academic Press (1982). Phys. B241 (1984) 333. Sierra: On a new class of integrable models. Phys. B6 (1992) 1939; q-Magnetism at roots of unity. to appear in J. A. LeClair: Non-local conserved currents: the lattice approach. LeClair: The quantum double in integrable quantum field theory. Preprint CLNS 92/1147 (1992). LeClair: Residual quantum symmetries of the restricted sine-Gordon theories. Phys. B340 (1990) 721. Lett. Phys. 142 (1991) 99. Lett. 206B (1988) 412. Pilch: Free field realization of WZNWmodels; BRST complex and its quantum group structure.

Spin chains at roots of unity To finish this brief history we would like to make some comments in connection to recent progress by De Concini and Kac [DK] in the study of the representation theory of quantum groups at roots of unity (see also [DKP]. As we have already mentioned the quantum deformation parameter of the quantum group associated to a RCFT is a root of unity, for example q = e'lfi/k+2 for a SU(2) WZW model. This implies that strictely speaking the quantum group associated to this RCFT should be Uq (SU(2)) moded out by the central Hopf which appears when q is a root of unity [McSo].

Gervais: The quantum group structure of 2D gravity and minimal models. l30 (1990) 257. Phys. B224 (1983) 329 ; Novel triangle relation and absence of tachyons in Liouville string field theory. Phys. B238 (1984) 125. Sierra: New R-matrices associated with finite dimensional representations of Uq (sl(2)) at roots of unity. Lett. 265B (1991) 95. Lett. 240B (1990) 149. Sierra: The quantum symmetry of rational conformal field theories. Phys. B352 (1991) 791. Sierra: Contour representation of quantum groups in RCFT's.

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