idea is to take into account the connection of every block with the rest of the system when choosing the states which survive the truncation procedure. The standard prescription is to choose the lowest energy states of the block Hamiltonian. Instead, in the DMRG method one replaces the block Hamiltonian by a block density matrix and chooses the eigenstates of this matrix with the highest eigenvalues. The density matrix is constructed out of the ground state of a superblock which contains the desired block.

In these notes we want to explain another RG method, the qRG method, which uses the concept of quantum groups. This mathematical notion emerged in the study of integrable systems and it has been applied to conformal field theory, invariants of knots and manifolds, etc. [@drinfeld-jimbo], [@germancesar]. The new application of quantum groups that we envisage has been partially motivated by the aforementioned work of White, Noack and collaborators and it is probably related to it. This relation is suggested by the fact that quantum groups describe symmetries in the presence of non-trivial boundary conditions. The typical example to understand this property of quantum groups is given by the 1D Heisenberg-Ising model with anisotropic parameter $\Delta 