Generalization of the Gell–Mann Decontraction Formula for sl(n, $\mathbb{R}$) and Its Applications in Affine Gravity

Abstract

The Gell–Mann Lie algebra decontraction formula was proposed as an inverse to the Inonu–Wigner contraction formula. We considered recently this formula in the content of the special linear algebras sl(n), of an arbitrary dimension. In the case of these algebras, the Gell–Mann formula is not valid generally, and holds only for some particular algebra representations. We constructed a generalization of the formula that is valid for an arbitrary irreducible representation of the sl(n) algebra. The generalization allows us to explicitly write down, in a closed form, all matrix elements of the algebra operators for an arbitrary irreducible representation, irrespectively whether it is tensorial or spinorial, finite or infinite dimensional, with or without multiplicity, unitary or nonunitary. The matrix elements are given in the basis of the Spin(n) subgroup of the corresponding SL(n, R) covering group, thus covering the most often cases of physical interest. The generalized Gell–Mann formula is presented, and as an illustration some details of its applications in the Gauge Affine theory of gravity with spinorial and tensorial matter manifields are given.