Projection operators for simple lie groups

R. M. Asherova ; Yu. F. Smirnov ; V. N. Tolstoy

Teoretičeskaâ i matematičeskaâ fizika, Tome 15 (1973) no. 1, pp. 107-119

Résumé

The solution of many problems of nuclear theory reduces to projecting wave functions $\psi$ that are not eigenfunctions of the integrals of motion $\Lambda$ onto the eigenfunetion space of these operators $\Lambda$. For this projection one requires projection operators for the groups $SU(n)$, $SO(n)$, and other simple Lie groups. In the present paper a general scheme is proposed, for an arbitrary simple Lie group $G(l)$ of rank $l$, for constructing raising and lowering operators $\mathscr F_{+}$ and $\mathscr F_{-}$, which, together with the previously obtained operators $P^{[f]}$, form cornplete projection operators for the given group. We are concerned with bases of irreducible representations of $G(l)$ which are such that they correspond to restriction to a chain of regularly imbedded subgroups $G(l)\supset G(g)\supset\dots\supset G(s)\supset\dots\supset G(t)$. As an example of a concrete realization of the scheme the lowering operators $\mathscr F_{-}$ are obtained for the canonical Gel'fand–Tseitlin basis for the group $U(n)$. The matrix elements of the generators of the group $U(n)$ are obtained in this basis.