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 TMF, 1971, Volume 8, Number 2, Pages 255–271 (Mi tmf4403)

Projection operators for simple lie groups

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

Abstract: The solution of many problems in nuclear theory and elementary particle physics amounts to decomposing the reducible representations of the symmetry groups of quantum mechanical systems into irreducible components. To carry out this decomposition, projection operators are needed. In the present paper we have constructed, for all simple compact Lie groups $G(l)$ of the rank $l$ (both classical and exceptional), operators which project the arbitrary vector with the weight $f=(f_1,f_2,…,f_l)$ onto the highest weight vector of the irreducible representation $D^{[f]}$ of the group $G(l)$. The projection operators are represented in the form of series composed of powers of the infinitesimal operators, which makes them convenient for the solution of particular problems concerning the decomposition of reducible representations into irreducible components. The structure of the projection operators is given for all simple compact Lie groups by similar formulas.

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English version:
Theoretical and Mathematical Physics, 1971, 8:2, 813–825

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Citation: R. M. Asherova, Yu. F. Smirnov, V. N. Tolstoy, “Projection operators for simple lie groups”, TMF, 8:2 (1971), 255–271; Theoret. and Math. Phys., 8:2 (1971), 813–825

Citation in format AMSBIB
\Bibitem{AshSmiTol71} \by R.~M.~Asherova, Yu.~F.~Smirnov, V.~N.~Tolstoy \paper Projection operators for simple lie groups \jour TMF \yr 1971 \vol 8 \issue 2 \pages 255--271 \mathnet{http://mi.mathnet.ru/tmf4403} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=475353} \zmath{https://zbmath.org/?q=an:0223.22019} \transl \jour Theoret. and Math. Phys. \yr 1971 \vol 8 \issue 2 \pages 813--825 \crossref{https://doi.org/10.1007/BF01038003} 

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This publication is cited in the following articles:
1. R. M. Asherova, Yu. F. Smirnov, V. N. Tolstoy, “Projection operators for simple lie groups”, Theoret. and Math. Phys., 15:1 (1973), 392–401
2. A. N. Leznov, M. V. Saveliev, “Projection operators for semisimple Lie groups”, Theoret. and Math. Phys., 31:2 (1977), 456–457
3. D. P. Zhelobenko, “Extremal projectors and generalized Mickelsson algebras over reductive Lie algebras”, Math. USSR-Izv., 33:1 (1989), 85–100
4. D. V. Yur'ev, “Topics in isotopic pairs and their representations. II. A general supercase”, Theoret. and Math. Phys., 111:1 (1997), 511–518
5. D. P. Zhelobenko, “Universal Verma modules and $W$-resolvents over Kač–Moody algebras”, Theoret. and Math. Phys., 122:3 (2000), 278–297
6. Yu. A. Neretin, S. M. Khoroshkin, “Mathematical works of D. P. Zhelobenko”, Russian Math. Surveys, 64:1 (2009), 187–198
7. A. I. Mudrov, “Regularization of Mickelsson generators for nonexceptional quantum groups”, Theoret. and Math. Phys., 192:2 (2017), 1205–1217
8. Andrey I. Mudrov, “Contravariant Form on Tensor Product of Highest Weight Modules”, SIGMA, 15 (2019), 026, 10 pp.
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