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

Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur’s Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna...

A bstract Nonrelativistic string theory is a self-contained corner of theory, with its spectrum enjoying Galilean-invariant dispersion relation. This unitary and ultraviolet complete, can be studied from first principles. In these notes, we focus on the bosonic closed sector. curved spacetime, nonrelativistic defined by renormalizable quantum nonlinear sigma model in background fields, followin...

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...