This paper is devoted to deformation theory of graded Lie algebras over Z or Zl with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artinian local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations...

We first introduce 20 weak forms of continuity, which are closely related to 5 known weak forms of continuity. Then we classify them into 9 groups and give 12 decompositions of continuity. 1. Introduction. Continuity is one of the most important concepts in mathematics. In order to find deep properties of continuity, many weak forms of continuity were introduced in the literature. For instance,...