Integrability of the one-dimensional Hubbard model and of the factorised
scattering problem encountered on the worldsheet of AdS strings can be
expressed in terms of a peculiar quantum algebra. In this article, we derive
the classical limit of these algebraic integrable structures based on
established results for the exceptional simple Lie superalgebra d(2,1;epsilon)
along with standard sl(2) which form supersymmetric isometries on 3D AdS space.
The two major steps in this construction consist in the contraction to a 3D
Poincar\’e superalgebra and a certain reduction to a deformation of the u(2|2)
superalgebra. We apply these steps to the integrable structure and obtain the
desired Lie bialgebras with suitable classical r-matrices of rational and
trigonometric kind. We illustrate our findings in terms of representations for
on-shell fields on AdS and flat space.