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 TMF, 1973, Volume 15, Number 1, Pages 107–119 (Mi tmf3644)

Projection operators for simple lie groups

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

Abstract: 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 $G(l)\supset G(g)\supset…\supset G(s)\supset…\supset G(t)$. 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.

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English version:
Theoretical and Mathematical Physics, 1973, 15:1, 392–401

Bibliographic databases:

Citation: R. M. Asherova, Yu. F. Smirnov, V. N. Tolstoy, “Projection operators for simple lie groups”, TMF, 15:1 (1973), 107–119; Theoret. and Math. Phys., 15:1 (1973), 392–401

Citation in format AMSBIB
\Bibitem{AshSmiTol73}
\by R.~M.~Asherova, Yu.~F.~Smirnov, V.~N.~Tolstoy
\paper Projection operators for simple lie groups
\jour TMF
\yr 1973
\vol 15
\issue 1
\pages 107--119
\mathnet{http://mi.mathnet.ru/tmf3644}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=475354}
\zmath{https://zbmath.org/?q=an:0253.22008}
\transl
\jour Theoret. and Math. Phys.
\yr 1973
\vol 15
\issue 1
\pages 392--401
\crossref{https://doi.org/10.1007/BF01028268}

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This publication is cited in the following articles:
1. Yu. A. Neretin, S. M. Khoroshkin, “Mathematical works of D. P. Zhelobenko”, Russian Math. Surveys, 64:1 (2009), 187–198
2. Raisa M. Asherova, Čestmír Burdík, Miloslav Havlíček, Yuri F. Smirnov, Valeriy N. Tolstoy, “$q$-Analog of Gelfand–Graev Basis for the Noncompact Quantum Algebra $U_q(u(n,1))$”, SIGMA, 6 (2010), 010, 13 pp.
3. O. V. Ogievetskii, S. M. Khoroshkin, “Diagonal Reduction Algebras of $\mathfrak{gl}$ Type”, Funct. Anal. Appl., 44:3 (2010), 182–198
4. Kibler M.R., “In memoriam of two distinguished participants of the Bregenz Symmetries in Science Symposia: Marcos Moshinsky and Yurii Fedorovich Smirnov”, Symmetries in Science XIV, Journal of Physics Conference Series, 237, 2010
5. Sergei Khoroshkin, Oleg Ogievetsky, “Structure Constants of Diagonal Reduction Algebras of $\mathfrak{gl}$ Type”, SIGMA, 7 (2011), 064, 34 pp.
6. A. I. Mudrov, “Regularization of Mickelsson generators for nonexceptional quantum groups”, Theoret. and Math. Phys., 192:2 (2017), 1205–1217
7. O. V. Ogievetskii, B. Herlemont, “Rings of $\mathbf h$-deformed differential operators”, Theoret. and Math. Phys., 192:2 (2017), 1218–1229
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