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Journal of Mathematical Physics / Volume 23 / Issue 1

J. Math. Phys. 23, 8 (1982); doi:10.1063/1.525212 (13 pages)

Modified fourth‐order Casimir invariants and indices for simple Lie algebras

Susumu Okubo

Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627

The fourth‐order indices for Lie algebras have been defined and studied by Patera, Sharp, and Winternitz. We show that it may be more convenient to modify the original definition and that the modified fourth‐order indices are intimately related to eigenvalues of symmetrized fourth‐order Casimir invariants. Explicit expressions for these quantities are given and we also find a quartic trace identity involving the generic element of these Lie algebras. We discuss the triality principle for the Lie algebra D4 in connection with identical vanishing of the modified fourth‐order index for this algebra.

KEYWORDS and PACS

Keywords

ALGEBRAS, EIGENVALUES, CASIMIR OPERATORS, LIE GROUPS

PACS

ARTICLE DATA

Permalink
http://link.aip.org/link/doi/10.1063/1.525212

PUBLICATION DATA

ISSN:

0022-2488 (print)  
1089-7658 (online)

For access to fully linked references, you need to log in.
    S. Okubo, J. Math. Phys. 20, 586 (1979JMAPAQ000020000004000586000001). This paper will be referred to as (I).

    J. Patera, R. T. Sharp, and P. Winternitz, J. Math. Phys. 17, 1972 (1976JMAPAQ000017000011001972000001),, Erratum 18, 1519 (1977JMAPAQ000018000007001519000001). We have modified their notation I(2p) to I2p(rho) since we reserve Ip(rho) for eigenvalues of pth-order Casimir invariants.

    S. Okubo, Phys. Rev. D 16, 3528 (1977).

    J. Banks and H. Georgi, Phys. Rev. D 14, 1159 (1976).

    Y. Tosa and S. Okubo, Phys. Rev. D 23, 3058 (1981).

    S. Okubo, Phys. Rev. D 18, 3792 (1978).

    M. X. F. Wong and H. Y. Yeh, J. Math. Phys. 16, 1239 (1975JMAPAQ000016000006001239000001).

    C. O. Nwachuku and M. A. Rashid, J. Math. Phys. 17, 1611 (1976JMAPAQ000017000008001611000001).

    S. Okubo, J. Math. Phys. 18, 2382 (1977JMAPAQ000018000012002382000001).

    S. A. Edwards, J. Math. Phys. 19, 164 (1978JMAPAQ000019000001000164000001).

    S. Okubo, Phys. Rev. C 10, 2048 (1974).

    For another way of formulating the triality and for its connection to the octonion algebra, see M. Günaydin and F. Gürsey, J. Math. Phys. 14, 1651 (1973JMAPAQ000014000011001651000001).


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