Abstract
We consider products of structure constants of a finite-dimensional compact simple Lie algebra, in which all indices except a few are contracted in pairs. We prove that such a product is zero if only one index is free, is proportional to the Cartan-Killing tensor if two indices are free and is proportional to a structure constant itself if three indices are free. For SU(n),n≧3 we also consider products of usuald (related to the anti-commutator) and structure constantsf. The results for one and two free indices are still valid. For three free indices the product is proportional to either anf or ad according to whether the number off's in the product is odd or even.
Similar content being viewed by others
References
Taylor, J.C.: Ward identities and charge renormalization of the Yang-Mills-field. Nucl. Phys. B33, 436 (1971)
Slavnov, A.: Ward identities in gauge theories. Theor. Math. Phys.10, 99 (1972)
Dhar, A., Gupta, V.: In preparation (1982)
These are the symmetric and asymmetric momentum subtraction schemes discussed In: Dhar, A., Gupta, V.: Pramana17, 469 (1981)
See for example, Hamermesh, M.: Group theory. Reading, MA: Addison-Wesley 1962, Sect. 5.6
Cartan, E.: Thesis (1894), reprinted in Oeuvres Complètes. Paris: Gauthier-Villars 1962 Part 1, Vol. 1;
Weyl, H.: Math. Z.23, 271 (1925);24, 328 (1926)
Mehta, M.L.: Classification of irreducible unitary representations of compact simple Lie groups. I. J. Math. Phys.7, 1824 (1966)
Mehta, M.L., Srivastava, P.K.: Classification of irreducible unitary representations of compact simple Lie groups. II. J. Math. Phys.7, 1833 (1966)
See for example, Macfarlane, A.J., Sudbery, A., Weisz, P.H.: On Gell-Mann's λ-matrices,d- andf-tensors, octets, and parametrizations of SU(3). Commun. Math. Phys.11, 77 (1968), for further useful relations involvingf andd
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
Rights and permissions
About this article
Cite this article
Metha, M.L., Normand, J.M. & Gupta, V. A property of the structure constants of finite dimensional compact simple Lie algebras. Commun.Math. Phys. 90, 69–78 (1983). https://doi.org/10.1007/BF01209387
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01209387