Lattices in potentially semi-stable representations and weak $(\phi ,\widehat{G})$-modules

Let $p$ be a prime number and $r$ a non-negative integer. In this paper, we prove that there exists an anti-equivalence between the category of weak $(\phi ,\widehat{G})$-modules of height $r$ and a certain subcategory of the category of Galois stable lattices in potentially semi-stable $p$-adic representations with Hodge-Tate weights in $[0,r]$. This gives an answer to a Tong Liu’s question about the essential image of a functor on weak $(\phi ,\widehat{G})$-modules. For a proof, following Liu’s methods, we construct linear algebraic data which classify lattices in potentially semi-stable representations.