The quantum double of the Yangian of the Lie superalgebra A(m, n) and computation of the universal R-matrix

On Drinfeld’s second realization of the AdS/CFT su(2|2) Yangian

“…In this section we summarize the definitions for the Yangian Y(g) of a simple Lie algebra g in Drinfeld's first and second realization and give the isomorphism between the two [21,28]. The generalisation to simple Lie superalgebras is straightforward (see for instance [23]), however, for simplicity, we refrain from giving the needed modifications in this section. The reader is referred to the standard literature (see for example [17,29]) for a thorough treatment of this subject.…”

“…Hence, it is natural to expect such S-matrix to be the representation of the universal R-matrix of a Yangian double [21], somehow modified by a twist in order to incorporate the braiding elements of [9,10]. Universal R-matrices for Yangians based on simple Lie (super)algebras have been derived in [22,23], but a main ingredient in the derivation, the existence of a non-degenerate invariant bilinear form, is missing for psu(2|2) ⋉ R 3 . One can in principle remove this degeneracy by adjoing the external sl(2) automorphisms to psu(2|2)⋉R 3 , but they have no finite dimensional representation, and they cannot directly be seen to be symmetries of the S-matrix.…”

Drinfeld Yangian of the queer Lie superalgebra. I

“…There was also obtained his description in terms of the current system of generators and defining relations (a new system of generators in the terminology of V. Drinfel'd, [5]). It should be noted that the Yangian of a queer Lie superalgebra is defined as the deformation of a Lie bisuperalgebra of twisted polynomial currents in which the structure of a Lie superalgebra is given by a rational r-matrix (see [20]).…”

Yangian of the Strange Lie Superalgebra of Qn-1 Type, Drinfel'd Approach

“…The RFT approach is used in the paper [14] (see also [6,7,8]) to define the Yangian of basic Lie superalgebra, while the Drinfel'd's one is used in papers [10,11,13]. We cannot use directly Drinfel'd approach for defining the Yangian of strange Lie superalgebra [16] as this Lie superalgebra does not have nonzero invariant bilinear forms.…”

“…Further, in [11,13] the quantum double of the Yangian of the Lie superalgebra A(m, n) is described, and the multiplicative formulas for the universal R-matrices (for both quantum double of the Yangian and Yangian) are obtained. This paper is the consequence of [10,12] and extension of some of their results on the Yangian of the "strange" Lie superalgebra.…”