Theoretical and Mathematical Physics, Vol. 124, No. 2, 2000 FACTORIZATION OF THE UNIVERSAL T~-MATRIX FOR Uq(slu) ~ J. Ding, 2 S. Pakuliak, a"~ and S. Khoroshkin 4 The factorization of the universal ~-matrix correspondi~g to the so-cMled Drinfeld Hopf structure is described in the example of the quantum affine algebra Uq(sl2). As a result of the factorization procedure, we deduce certain differential equations on the factors of the universal 7"~-matrix that allow uniquely constructing these factors in the integral form. Contents Appendix: Pairing calculations 1. Introduction ..................................................... 1007 2. Prelitninaries and main results ......................................... 1009 2.1. Two descriptions of the quantum affine algebra Uq(sl2) ..................... 1009 2.2. Main results ................................................. 1012 3. Uq(,~,2) with the "Drinfeld" coproduct ..................................... 1014 3.1. The universal 74-matrix in a multiplicative form .......................... 1014 3.2. Integral representation for the element ~ and the fimdamental differential equation . . . 1016 3.3. The pairing tensor as a formM integral ................................ 1017 4. Factorlzation of the universal T4-matrix ................................... 1019 4.1. The biorthogonal decompositions of Hopf algebras ......................... 1019 4.2. Application to the algebra Uq(sl2) ................................... 1022 4.3. Differential equations for the elements T4+,~=(r) ........................... 1023 4.4. Projections of composed currents and screening operators .................... 1025 5. Factorization of the formal paMng tensor .................................. 1028 5.1. Another form of tile differential equations and the combinatorial identity .......... 1028 5.2. Some examples of calculations and vanishing of the cross terms in tile integrals ...... 1031 ........................................... 1033 1. Introduction Many group theory methods for investigating quantum integrable models originated from quantum group theory. The quantum group theory based on the quantum inverse scattering method [1] was described in pioneer papers [2, 3] as the Hopf algebra deformation of the universal enveloping algebras of contrag'radient Lie algebras. In most applications, the quantum groups as Hopf algebras appear together with R-matrices, either in the form of numerical matrices or L-operators or universal 7~-matrices. The latter are the elements acting in the completed square of the corresponding Hopf algebras satisfying certain conditions. These 1[This article was written at the request of the Editorial lBoard.] 2Department of Ma.thematical Sciences, Uuivat:~ityof Cincinnati, USA, e-ma.il: dintai.Ding(6)math.uc.edu. 3Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna., Moscow Oblast, Russia; Bogoliubov Institute fin"The<)- retk'al Physics, Kiev, Ukraine, e-mail: l)akuliak~.0thsunl.jinr.ru. "llnstitute for Theoretical and Experimental Physics, Moscow, [{ussia, e-mail: khorg~-heron.itep.ru. 1u from 1~oreticheskaya i Matematicheskaya Fizika, ~v~)l. 124, No. 2, pp. 179 214, August, 2000. Orig- inal article submitted March 30, 1999. 0040-5779/00/1242-1007525.00 @ 2000 Kluwer Acadenfic/Plenun~ Publishers 1007