Casimir operators for semisimple Lie groups

A. M. Perelomov ; V. S. Popov

Izvestiya. Mathematics , Tome 2 (1968) no. 6, pp. 1313-1335.

Résumé

A simple method is developed for computing the eigenvalues of the invariant operators (the so-called Casimir operators) $\widehat C_p$ of arbitrary order $p$ for semisimple Lie groups. The resulting formulas (52) and (55) are applicable for the case that among the irreducible representations of the given group there is at least one representation with a simple spectrum – in particular, for all the classical groups, as well as the groups $G_2$, $E_6$ and $E_7$. An expression is found (see (75)) for the generating function of the Casimir operators in the case of the classical groups.

Bibliographie

[1] Casimir H., “Über die Konstmction einer zu den irreduzibelen Darstelhmgen halbeinfacher Kontinuiriicher Gruppen gehörigen Differenzialgleichung”, Proc. Nider. Akad. van Wet., 34 (1931), 844–846 | Zbl

[2] Casimir H., van der Waerden B. L., “Algebraischetf Beweis der vollständiger Reduzibilität der Darstellungen halbeinfacher Liescher Gruppen”, Math. Ann., 111 (1935), 1–12 | DOI | MR | Zbl

[3] Whitehead J. H. C., “Certain equations in the algebra of an infinitesimal semisimple group”, Quart. J. Math., 8 (1937), 220–237 | DOI | Zbl

[4] Gelfand I. M., “Tsentr infinitezimalnogo gruppovogo koltsa”, Mat. sb., 26 (1950), 103–112 | MR

[5] Chevalley C., “Invariants of finite groups generated by reflections”, Amer. J. Math., 77 (1955), 778–782 | DOI | MR | Zbl

[6] Racah G., Group theory and spectroscopy, Lecture notes, Princeton, 1951 | Zbl

[7] Racah G., “Sulla caraterizzazione delle rappresentazioni irriducibili dei gruppi semisemplici di Lie”, Rend. Lincei, 8 (1950), 108–112 | MR | Zbl

[8] Bepezin F. A., “Operatory Laplasa na poluprostykh gruppakh Li”, Tr. Mosk. Matem. o-va, 6 (1957), 371–463

[9] Racah G., “Theory of complex spectra. IV”, Phys. Rev., 76 (1949), 1352–1365 ; L. C. Biedenharn, H. van Dam (eds.), “Nuclear levels and Casimir operators”, перепечатано в сб.:, Farkas Memorial Volume, Jerusalem, 1952, 294 | DOI | Zbl

[10] Biedenharn L. C., “Nuclear spectroscopy”, Lectures in theoretical physics, v. 5, New York, 1963, 258–421

[11] Pais A., “Dynamical symmetry in particle physics”, Revs. Mod. Phys., 38 (1966), 215–255 | DOI

[12] Zhelobenko D. P., Lektsii po teorii grupp Li, Dubna, 1965

[13] Umezawa M., “Invariant quantities in semisimple groups. I”, Nucl. Phys., 48 (1963), 111–120 ; “II”, 53 (1964), 54–64 ; “III”, 57 (1964), 65–88 | DOI | MR | DOI | MR | Zbl | DOI | MR

[14] Umezawa M., “Invariant quantities in simple groups $B_\omega$, $C_\omega$ and $D_\omega$. I”, Proc. Neder. Akad. van Wet(B), 69 (1966), 579 ; “II”, 592 ; “III”, 607 ; “IV”, 620 | MR | Zbl

[15] Micu M., “Construction of invariants for simple Lie groups”, Nucl. Phys., 60 (1964), 353–362 | DOI | MR | Zbl

[16] Biedenharn L. C., “On the representations of the semisimple Lie groups. I. The explicit construction of invariants for the unimodular unitary group in $N$ dimensions”, J. Math. Phys., 4 (1963), 436–445 | DOI | MR | Zbl

[17] Baird G. E., Biedenharn L. C., “On the representations of the semisimple Lie groups. II”, J. Math. Phys., 4 (1963), 1449–1466 | DOI | MR

[18] Gruber B., O'Kaifeartaigh L., “$S$-theorem and construction of the invariants of the semisimple compact Lie algebras”, J. Math. Phys., 5 (1964), 1796–1804 | DOI | MR | Zbl

[19] Harish-Chandra, “On some applications of the universal enveloping algebra of semisimple Lie algebra”, Trans. Amer. Math. Soc., 70 (1951), 28–96 | DOI | MR | Zbl

[20] Berezin F. A., “Neskolko zamechanii ob assotsiativnoi obolochke algebry Li”, Funktsionalnyi analiz i ego prilozheniya, 1:2 (1967), 1–14 | MR | Zbl

[21] Perelomov A. M., Popov V. S., “Operatory Kazimira dlya unitarnoi gruppy”, Pisma v ZhETF, 1:6 (1965), 15–18 | MR

[22] Perelomov A. M., Popov V. S., “Operatory Kazimira dlya grupp $U(n)$ i $SU(n)$”, Yadernaya fizika, 3 (1966), 924–931 | MR

[23] Perelomov A. M., Popov V. S., “Operatory Kazimira dlya klassicheskikh grupp”, Dokl. AN SSSR, 174 (1967), 287–290 | MR | Zbl

[24] Zhelobenko D. P., “Klassicheskie gruppy. Spektralnyi analiz konechnomernykh predstavlenii”, Uspekhi matem. nauk, 17:1 (1962), 27–120 | MR

[25] Behrends R. E., Dreitlein J., Fronsdal C., Lee B. W., “Simple groups and strong interaction symmetries”, Revs. Mod. Phys., 34 (1962), 5 | MR

[26] Teoriya algebr Li. Topologiya grupp Li, IL, M., 1962

[27] Dynkin E. V., “Maksimalnye podgruppy klassicheskikh grupp”, Tr. Mosk. Matem. o-va, 1 (1952), 39–166 | MR | Zbl

[28] Eiler L., Vvedenie v analiz beskonechnykh, t. I, Fizmatgiz, M., 1961

[29] Higher transcendental functions, v. 3, Mc Graw-Hill Book Co., N. Y., 1955

[30] Lomont J. S., Applications of finite groups, Academic press, N. Y., 1959 | MR | Zbl