Weak Faddeev–Takhtajan–Volkov algebras. Lattice $W_n$ algebras

In this paper, we will start by deliberating at our project's historical general view and then we will try to construct a new Poisson bracket on our simplest example $sl_2$ and then we will try to give a universal construction based on our universal variables and then will try to construct lattice $W_2$ algebras which will play a key role in our other constructions on lattice $W_3$ algebras and finally we will try to find the only nontrivial dependent generator of our lattice $W_4$ algebras and so on for lattice $W_n$ algebras.