  This article is made up with two parts. In the first part, we generalize to
the case when objects are faithfully flat over the ground ring, the full
equivalence between the notions of Hopf-Galois extension and Hopf-Galois
system. In the second part of the article, we show that the Kashiwara algebras
introduced by Kashiwara in his study of crystal bases are Hopf-Galois objects.
