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Journal of Mathematical Physics / Volume 20 / Issue 4

J. Math. Phys. 20, 586 (1979); http://dx.doi.org/10.1063/1.524127 (8 pages)

Quartic trace identity for exceptional Lie algebras

Susumu Okubo

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

Let X be a representation matrix of generic element x of a simple Lie algebra in generic irreducible representation {λ} of the Lie algebra. Then, for all exceptional Lie algebras as well as A1 and A2, we can prove the validity of a quartic trace identity Tr(X4) =K (λ)[Tr(X2)]2, where the constant K (λ) depends only upon the irreducible representation {λ}, and its explicit form is calculated. Some applications of second and fourth order indices have also been discussed.

KEYWORDS and PACS

Keywords

LIE GROUPS, ALGEBRA, IRREDUCIBLE REPRESENTATIONS, EQUATIONS, INVARIANCE PRINCIPLES

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.524127

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. 18, 2382 (1977)JMAPAQ000018000012002382000001. This paper will herafter be referred to as (I).

    P. Cvitanović, private communication, where he uses a technique developed in P. Cvitanović, Phys. Rev. D 14, 1536 (1976) and University of Oxford Report 40/77 (1977) (unpublished). The author would like to express his gratitude to Dr. Cvitanović for informing him of these results.

    J. D. Louck and L. C. Biedenharn, J. Math. Phys. 11, 2368 (1970)JMAPAQ000011000008002368000001;, 14, 1336 (1973)JMAPAQ000014000010001336000001;, L. C. Biedenharn and J. D. Louck, J. Math. Phys. 13, 1985 (1972)JMAPAQ000013000012001985000001;, L. C. Biedenharn, J. D. Louck, F. Chacón, and M. Ciftan, J. Math. Phys. 13, 1957 (1972)JMAPAQ000013000012001957000001.

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

    J. Patera, R. T. Sharp, and P. Winternitz, J. Math. Phys. 17, 1972 (1976)JMAPAQ000017000011001972000001;, 18, 1519(Erratum) (1977)JMAPAQ000018000007001519000001. Note that our notation l2P(lambda) corresponds to I(2p)(lambda) of theirs.


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