ar X iv : q - a lg / 9 70 30 22 v 1 1 2 M ar 1 99 7 Drinfel ′ d Realization of Quantum Affine

We obtain Drinfel ′ d's realization of quantum affine superalgebra U q (gl(1|1)) based on the super version of RS construction method and Gauss decomposition. How to construct quantum algebras (including quantum affine algebras and Yangians) is a important problem in mathematics physics. After Drinfel ′ d [1] and Jimbo [3, 4] independently discovered that the universal enveloping algebra U(g) of any simple Lie algebras or Kac-Moody algebras admits a Hopf algebraic structure and a certain q-deformation, Drinfel ′ d [2] gave his second definition or realization of quantum affine algebras U q (g) and Yangians. From views of quantum inverse scattering method, Faddeev, Reshetikhin and Takhtajan (FRT) [5] gave another realization of U q (g) by means of a solution of Yang-Baxter equation (YBE). The FRT method has a much more direct physics meaning and can be extended to quantum loop algebra U q (g ⊗ [t, t −1 ]) when the solution of YBE be dependent on spectrum parameter. Laterly, Reshetikhin and Semenov-Tian-Shansky (RS) [6] used the exact affine anologue of FRT method to obtain a realization of quantum affine algebra U q (g). The explicit isomorphism between two realizations of quantum affine algebras U q (g) given by Drinfel ′ d and RS was established by Ding and Frenkel [7] using Gauss decomposition. However, for a non-standard solution (without spectrum parameter) of YBE, if we employ the usual FRT constructive method, we would obtain a peculiar quantum algebra whose classical limit is not a Lie superalgebra despite some of its relations (such as X 2 = Y 2 = 0) look like fermionic relations [8]. It was pointed out by Liao and Song [9] that, to deal with a would-be quantum Lie superalgebra from this nonstandard solution of YBE, one must start from formulas (YBE and RLL relations etc.) appropriate for the super case at the very begining—super version of FRT method. We must use super version of FRT (RS) method to to construct quantum (affine) superalgebras [9, 10]. Recently, some attentions have been paid to the construction of the quantum affine superalgebras [11, 12]. In this paper, we will use the super affine version FRT method (or super RS method) to construct quantum affine superalgebra and use Gauss decomposition to get its Drinfel ′ d realization. We will focus on the simplest one: U q (gl(1|1)), and this method can be easily extended to general case.