Discrete and continuous graded contractions of Lie algebras and superalgebras

Grading preserving contractions of Lie algebras and superalgebras of any type over the complex number field are defined and studied. Such contractions fall naturally into two classes: the Wigner-Inonu-like continuous contractions and new discrete contractions. A general method is described for any Abelian grading semigroup and any Lie algebra or superalgebra admitting such a grading. All contractions preserving z 2- , z 3 -, and z 2 *z 2 -gradings are found. Examples of these gradings and contractions for the simple Lie algebra A 2 , affine Kac-Moody algebra A (1) and the simple superalgebra osp(2,1) are shown.