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

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