Abstract
We study degenerations of the Belavin R-matrices via the infinite dimensional operators defined by Shibukawa–Ueno. We define a two-parameter family of generalizations of the Shibukawa–Ueno R-operators. These operators have finite dimensional representations which include Belavin's R-matrices in the elliptic case, a two-parameter family of twisted affinized Cremmer–Gervais R-matrices in the trigonometric case, and a two-parameter family of twisted (affinized) generalized Jordanian R-matrices in the rational case. We find finite dimensional representations which are compatible with the elliptic to trigonometric and rational degeneration. We further show that certain members of the elliptic family of operators have no finite dimensional representations. These R-operators unify and generalize earlier constructions of Felder and Pasquier, Ding and Hodges, and the authors, and illuminate the extent to which the Cremmer–Gervais R-matrices (and their rational forms) are degenerations of Belavin's R-matrix.
Similar content being viewed by others
References
Antonov,A. Hasegawa,K.and Zabrodin,A.:On trigonometric intertwining vectors and non-dynamical R-matrix for the Ruijsenaars model,Nuclear Phys.B 503(1997), 747–770.
Belavin, A.A.:Dynamical symmetry of integrable quantum systems,Nuclear Phys.B 180(1981),189–200.
Cremmer,E.and Gervais, J.-L.:The quantum group structure associated with non-lin-early extended Virasoro algebras,Comm Math.Phys. 134(1990),619–632.
Ding,J.and Hodges, T.J.:The Yang-Baxter equation for operators on function elds, In:A.Pressley (ed.),Quantum Groups and Lie Theory,Cambridge University Press, 2001.
Endelman,R.and Hodges,T.J.:Generalized Jordanian R-matrices of Cremmer-Gerv-ais type,Lett.Math.Phys. 52(2000),225-237.
Felder,G.and Pasquier,V.:A simple construction of elliptic R-matrices,Lett.Math. Phys. 32(1994),167–171.
Gerstenhaber,M.and Giaquinto,A.:Boundary solutions of the classical Yang-Baxter equation,Lett.Math.Phys. 40(4)(1997),337–353
Gerstenhaber,M.and Giaquinto,A.:Boundary solutions of the quantum Yang-Baxter equation and solutions in three dimensions,Lett.Math.Phys. 44(1998), 131–141.
Hodges, T.J.:On the Cremmer Gervais quantizations of SL (n),Internat.Math.Res. Notices 10(1995),465–481.
Klimyk,A.and Schmüdgen,K.:Quantum Groups and their Representations,Springer New York,1997.
Shibukawa,Y.and Ueno,K.Completely Z-symmetric R matrix,Lett.Math.Phys. 25(1992),239–248.
Whittaker, E.T.and Watson, G.N.:A Course of Modern Analysis,Cambridge Univ. Press,1999.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Endelman, R., Hodges, T.J. Degenerations and Representations of Twisted Shibukawa–Ueno R-Operators. Letters in Mathematical Physics 68, 151–164 (2004). https://doi.org/10.1023/B:MATH.0000045551.15442.52
Issue Date:
DOI: https://doi.org/10.1023/B:MATH.0000045551.15442.52