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In this paper, the solutions of the Yang–Baxter equation both in the vertex model (R matrix) and in the face model (braiding matrix) are computed explicitly for the octet representation of the quantum Kac–Moody A(1)2 enveloping algebra where there is multiplicity greater than one in the decomposition of the coproduct 8x8. [ABSTRACT FROM AUTHOR]

The development of a Riemann–Hilbert (RH) problem approach to the representation of the Virasoro group introduced in our earlier paper [W. Li and B. Y. Hou, J. Math. Phys. 30, 1198 (1989)] is continued. The Rouche theorem is employed to prove that there exists a set of scalar functions in the complex τ plane, in terms of which the structure of the Virasoro group can be decided. The general form of the RH problem will also be presented and the uniqueness of its solution will be proven. For future use, a standard form of a matrix Fredholm equation equivalent to our RH problem will be deduced. This may help to establish in a future paper the existence of a solution of the RH problem. [ABSTRACT FROM AUTHOR]

The quantum and classical algebras connected with the Zn-symmetric elliptic solution of the Yang–Baxter equation are derived; their structure constants and the relations between the quantum algebra and the classical one are investigated in detail. Moreover, the trigonometric limit of these algebras is worked out. [ABSTRACT FROM AUTHOR]

In this paper, it is the intent to apply the Riemann–Hilbert transformation developed by Hauser and Ernst [J. Math. Phys. 21, 1126, 1418 (1980)] in providing a new representation of the Virasoro group. It is found that the Geroch group that acts on the solution space of the Einstein field equations is extended to the semidirect product of the Virasoro and Kac–Moody groups; also, the relationship between the infinitesimal transformation given previously [B. Y. Hou and W. Li, Lett. Math. Phys. 13, 1 (1987); J. Phys. A 20, L897 (1987); W. Li, Phys. Lett. A 129, 301 (1988)] and the infinitesimal Riemann–Hilbert transformation is pointed out. Finally, it is shown that the well-known Neugebauer–Backlund transformation can be derived from the Riemann–Hilbert transformation. [ABSTRACT FROM AUTHOR]

Using the natural connection form on a principal bundle P(M,G) and the bundle [ATOTHER]@FA[/ATOTHER]([ATOTHER]@FA[/ATOTHER]/G,G) a systematic derivation of the double-cohomological series constituted by the exterior differential d on space-time M and arbitrary, horizontal, and vertical variations in connection space is given. The relationship between these cohomologies and the family index theorem is clarified. The formalism is then used to analyze Abelian gauge structure inside non-Abelian gauge theory. The pertinent functional U(1) connection form, curvature form, and three-form ‘‘curvature’’ are identified and computed, and are related to the θ vacuum, anomalous commutation relation, and Jacobi identity, respectively. Some of the results differ from those obtained by Wu and Zee [Nucl. Phys. B 258, 157 (1985)] and Niemi and Semenoff [Phys. Rev. Lett. 55, 227 (1985)] and the results of this paperrecover theirs under certain conditions. Finally the generalization of the formalism to a nontrivial principal bundle by introduction of a fixed background connection form is discussed. [ABSTRACT FROM AUTHOR]

We study the elliptic C[sub n] and BC[sub n] Ruijsenaars–Schneider models which are elliptic generalization of systems given in previous paper by the present authors [Chen et al., J. Math. Phys. 41, 8132 (2000)]. The Lax pairs for these models are constructed by Hamiltonian reduction technology. We show that the spectral curves can be parametrized by the involutive integrals of motion for these models. Taking nonrelativistic limit and scaling limit, we verify that they lead to the systems corresponding to Calogero–Moser and Toda types. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

From the dynamical twisting of the classical r-matrix, we obtain a new Lax operator for the elliptic Ruijsenaars-Schneider model (cf. Ruijsenaars). The corresponding r-matrix is shown to be the classical Z[sub n]-symmetric elliptic r-matrix, which is the same as that obtained in the study of the nonrelativistic version--the A[sub n-1] Calogero-Moser model. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We study the C[sub n] and BC[sub n] Ruijsenaars-Schneider models with interaction potential of trigonometric and rational types. The Lax pairs for these models are constructed and the involutive Hamiltonians are also given. Taking a nonrelativistic limit, we also obtain the Lax pairs for the corresponding Calogero-Moser systems. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]