The Yangian double DY (A(m, n)) of the Lie superalgebra A(m, n) is described in terms of generators and defining relations. Normally ordered bases in the Yangian and its dual in the quantum double are introduced. We calculate the pairing between the elements of these bases and obtain a formula for the universal R-matrix of the Yangian double as well as a formula for the universal R-matrix (introduced by Drinfeld) of the Yangian.
1.Yangians of classical Lie superalgebras [10,11,8,9] have recently become a subject of study, along with Yangians of simple and reductive Lie algebras [1][2][3][4][5][6]. The Yangian of the Lie superalgebra A(m, n) was defined in [8] as a quantized polynomial current Lie bisuperalgebra with coalgebra structure given by the Yang r-matrix. In the same paper, the Yangian was described in terms of generators and relations which are an analog of the "new system of generators" introduced in [3], and the Poincaré-Birkhoff-Witt theorem and the theorem on the existence of a universal R-matrix were stated. The present paper is a continuation of [8], and its definitive result is an explicit formula for the universal R-matrix of the Yangian of A(m, n). The current Lie superalgebra A(m, n) [u] is the classical counterpart of the Yangian, and the classical r-matrix is the classical counterpart of the universal R-matrix. Therefore, in the quantum case it is also natural to set R(λ) = (id ⊗T λ )R, where R is the universal R-matrix of the Yangian double. Following this scheme, we introduce the Yangian double DY (A(m, n)) (the quantum double of the Yangian of A(m, n)) by describing it in terms of generators and relations. Here we follow [13], where the Yangian double was introduced for a simple Lie algebra. Next, we derive pairing formulas in DY (A(m, n)) and use them to obtain a formula for the universal R-matrix of the Yangian double. As was mentioned above, this formula is in turn used to find an expression for the universal R-matrix of the Yangian.We would like to note the distinction between the notions of universal R-matrices of the Yangian and its quantum double. Historically first, the notion of Yangian appeared and the existence of a universal R-matrix of the Yangian was proved (see [1,2,4]). But the problem of explicit computation of this R-matrix has long defied solution. In the mid-1990s, it became clear (see [13]) that computing the universal R-matrix of the Yangian double [4] (which, unlike the universal R-matrix of the Yangian, is a naturally defined object) is a sounder problem, and an explicit description of the Yangian double was given for simple Lie algebras. Since then, no one has returned to the problem of calculating the universal R-matrix of the Yangian. In the present paper, we show that the universal R-matrix of the Yangian can be obtained by applying a certain shift operator to the universal R-matrix of the quantum double and compute the former after preliminarily calculating the latter. We also note that the scheme for computing the universal R-matrix of the Yangian double is i...