On Lie algebras arising from p-adic representations in the imperfect residue field case

Let K   be a complete discrete valuation field of mixed characteristic (0,p)(0,p) with residue field kKkK such that [kK:kKp]=pd<∞. Let GKGK be the absolute Galois group of K   and ρ:GK→GLh(Qp)ρ:GK→GLh(Qp) a p  -adic representation. When kKkK is perfect, Shankar Sen described the Lie algebra of ρ(GK)ρ(GK) in terms of so-called Sen's operator Θ for ρ  . When kKkK may not be perfect, Olivier Brinon defined d+1d+1 operators Θ0,…,ΘdΘ0,…,Θd for ρ, which reduce to Sen's operator Θ   in the case of d=0d=0. In this paper, we describe the Lie algebra of ρ(GK)ρ(GK) in terms of Brinon's operators Θ0,…,ΘdΘ0,…,Θd, which is a generalization of Sen's result.