Contractions of invariants of Lie algebras with applications to classical inhomogeneous Lie algebras

Abstract

We study in full generality the contraction of invariants of finite-dimensional Lie algebras and give some illustrative examples. Our applications show that the contraction method is not only quite flexible but also has the advantage over other methods that independent invariants can be contracted into independent ones and that they appear in compact tensorial form. For our applications of physical interest, we can restrict ourselves to polynomial invariants (Casimir operators) and generalized Inönü–Wigner contractions. We show how to get all invariants of the inhomogeneous Lie algebras iso(p,q),iu(p,q),isl(n,R), and isp(2n,R) from those of the homogeneous Lie algebras. These invariants are all known, but our results for isl(n,R) and isp(2n,R) appear in a much more desirable form.