Universal R-matrix Of The Super Yangian Double DY (gl(1|1)) a Jin-fang Cai, bc Shi-kun Wang, a

Based on Drinfel ′ d realization of super Yangian Double DY (gl(1|1)), its pairing relations and universal R-matrix are given. By taking evaluation representation of universal R-matrix, another realization L ± (u) of DY (gl(1|1)) is obtained. These two realizations of DY (gl(1|1)) are related by the supersymmetric extension of Ding-Frenkel map. Yangian algebra was introduced by Drinfel ′ d[1, 2]. The quantum double of Yangian consists of Yangian itself and its dual with opposite comultiplication. There are three methods to define the Yangian and Yangian double: Drinfel ′ d-Jmbo [1, 3], Drinfel ′ d new realization [2] and RS approach [5] (or FRT approach [4] in the case of without center extension). The explicit isomorphism between Drinfel ′ d new realization and RS realization of Yangian double can be established through Gauss decomposition, the similar method used by Ding and Frenkel in the discussions of quantum Affine algebra. [6]. The property of Yangian double, such as quasi-triangular properties and equivalence of Drinfel ′ d and RS realization was studied well in some papers [7, 8, 9]. Although the Drinfel ′ d realization of super Yangian double [10, 11] was constructed by means of RS method and Gauss decomposition, the quasi-triangular property, such as universal R-matrix of super Yangian double has not been studied yet. In this paper, we find the Hopf pairing relations between super Yangian Y (gl(1|1)) and its dual, then construct the universal R-matrix of DY (gl(1|1)). By taking evaluation representation, we get the FRT realization of DY (gl(1|1)). Super Yangian double DY (gl(1|1)) is the Hopf algebra generated by elements