Discrete and continuous graded contractions of representations of Lie algebras

Simultaneous grading of Lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all Lie algebras admitting the chosen grading. The theory is completely different from the traditional ways of contracting representations. The graded contractions fall naturally into two classes: discrete and continuous ones related respectively to 2-cocycles and coboundaries of the grading group.