THE GENERAL STRUCTURE OF G-GRADED CONTRACTIONS OF LIE ALGEBRAS, II: THE CONTRACTED LIE ALGEBRA

Abstract

We continue our study of G-graded contractions γ of Lie algebras where G is an arbitrary finite Abelian group. We compare them with contractions, especially with respect to their usefulness in physics. (Note that the unfortunate terminology "graded contraction" is confusing since they are, by definition, not contractions.)We give a complete characterization of continuous G-graded contractions and note that they are equivalent to a proper subset of contractions. We study how the structure of the contracted Lie algebra Lγdepends on γ, and show that, for discrete graded contractions, applications in physics seem unlikely.Finally, with respect to applications to representations and invariants of Lie algebras, a comparison of graded contractions with contractions reveals the insurmountable defects of the graded contraction approach. In summary, our detailed analysis shows that graded contractions are clearly not useful in physics.