CONTRACTIONS, GENERALIZED INÖNÜ-WIGNER CONTRACTIONS AND DEFORMATIONS OF FINITE-DIMENSIONAL LIE ALGEBRAS

Abstract

We show that any contraction is equivalent to a generalized Inönü–Wigner contraction with integer exponents, thus solving a long-standing problem to find a "simple" class of contractions that can produce all possible contractions. These contractions are given by diagonal matrices of the form T (ε)jj = εnj, nj∈ ℤ. They are ideally suited for applications. Indeed, we use this result to show that contractions are inverse to analytic deformations, thus resolving another long-standing problem. To achieve reciprocity between contractions and deformations, we have extended the definition of contractions by dropping the requirement that T (0) = lim ε → 0T (ε) exists. We give an example which proves the necessity of this extended definition.