Hopf algebra structure of the AdS/CFTS-matrix

Abstract

We formulate the Hopf algebra underlying the su(2|2) worldsheet S-matrix of the AdS 5 × S 5 string in the AdS/CFT correspondence. For this we extend the previous construction in the su(1|2) subsector due to Janik to the full algebra by specifying the action of the coproduct and the antipode on the remaining generators. The nontriviality of the coproduct is determined by length-changing effects and results in an unusual central braiding. As an application we explicitly determine the antiparticle representation by means of the established antipode.

The Classical r-Matrix of AdS/CFT and its Lie Bialgebra Structure

“…Secondly, not all Lie brackets have a uniform level: there is mixing between the levels. In comparison to the Hopf algebra symmetries of the S-matrix [9,10,17] which use the three-dimensional universal central extension of psu(2|2), here we have only one central extension C. The three central elements of psu(2|2) R 3 clearly have no dual partners (automorphisms) which would be needed for the standard construction of the classical r-matrix. Conversely, our central element C can be paired with the u(1) automorphism B of u(2|2) as in the proposal of [29], and we can construct a classical r-matrix.…”

“…Here the modification of the summation bounds in (4.13) is crucial; without it some terms would remain. 10 …”

“…To achieve a quasi-cocommutative algebra, the central charges P 0 , K 0 , P 1 , K 1 must be identified with the braiding factor U and the central charge C 0 as follows [10,17] 2…”

“…Interestingly, the S-matrix is already fully fixed [7] up to a factor by demanding only invariance under the Lie algebra generators of psu(2|2) R 3 , without referring to an additional loop algebra or something similar. One only needs to introduce an additional braiding element [9,10] and identify the central charges and the braiding such that the central elements are all cocommutative. In that case, the S-matrix also depends on spectral parameters.…”

Quantum deformed magnon kinematics

“…The meaning of this kinematical symmetry must be understood from the structure of the Hopf algebra symmetry [8,9,10,5,11,12]. In [8] the existence of a central Hopf subalgebra was noticed and the spectrum of this center was proposed as the rapidity plane.…”