en تخصصي Special پژوهشي Research paper <div style="text-align: justify;">This paper is devoted to deformation theory of graded Lie algebras over Z or Z_lwith 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. In the second part, we use a version of Schlessinger criteria for functors on the<br> Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic deformations using this technique.</div> Arithemtic Lie algebras, Deforfation of Lie algebras, Schlessinger criteria. 19 32 http://ijmsi.ir/browse.php?a_code=A-10-2299-2&slc_lang=en&sid=1 Arash Rastegar rastegar1352@gmail.com 100319475328460010006 100319475328460010006 Yes Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran