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The semi-dynamical reflection equation: Solutions and structure matrices. (English) Zbl 1145.47060

The present paper adds knowledge in respect to discovering and classifying reflection matrices solution of the (semi)dynamical reflection equations introduced in [Z.Nagy, J.Avan and G.Rollet, Lett.Math.Phys.67, No.1, 1–11 (2004; Zbl 1058.81042)]. The first part of the paper concerns the possibility to extend the results for reflection matrices obtained in [J.Avan and G.Rollet, Ann.Henri Poincaré 7, No.7–8, 1463–1476 (2006; Zbl 1113.81066)] to the case with spectral parameter dependence. The authors consider rational non-constant Arutyunov–Chekhov–Frolov (ACF) structure matrices [G.E.Arutyunov, L.O.Cekhov and S.A.Frolov, Commun.Math.Phys.192, 405–432 (1998; Zbl 0973.81044)] and find that only two of the four basic sets of solutions of the constant case are extendible once a meromorphic ansatz for reflection matrices is imposed. The corresponding unique extensions are given.
A second purpose of the paper is to provide explicit realizations of the parametrization procedure proposed in [J.Avan and G.Rollet, J. Phys.A, Math.Theor.40, No.11, 2709–2731 (2007; Zbl 1111.81083)] for all elements of the semidynamical reflection equations and, incidentally, to investigate the possibility that the procedure may be helpful in shedding light on the properties of solutions of the equations. The authors remark that application of the procedure making use of the ACF non-constant matrices yields an apparently new R-matrix which satisfies a shifted Yang–Baxter equation. Further, they show that the the existence of distinct parametrization for the D-type structure matrix results in different sets of structure for the semidynamical reflection equation, and thus gives rise to a new reflection matrix solving the equation. In doing so, alternative parametrizations provided in [A.Antonov, K.Hasegawa and A.Zabrodin, Nucl.Phys., B 503, No.3, 747–770 (1997; Zbl 0933.82017)] are used.

MSC:

47N30 Applications of operator theory in probability theory and statistics
47N50 Applications of operator theory in the physical sciences
47L90 Applications of operator algebras to the sciences
81R12 Groups and algebras in quantum theory and relations with integrable systems
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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