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On tensor operators and characteristic identities for semi-simple Lie algebras

Published online by Cambridge University Press:  17 February 2009

M. D. Gould
M. D. Gould
Affiliation:
Department of Mathematical Physics, University of Adelaide, Adelaide, South Australia, 5001
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Abstract

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Tensor identities for finite dimensional representations of arbitrary semi-simple Lie algebras are derived and are applied to the construction of left-projection operators which project out the shift components of tensor operators from the left. The corresponding adjoint identities are also derived and are used for the construction of right-projection operators. It is also shown that, on a finite dimensional irreducible representation, these identities may be considerably reduced. Commutation relations between the shift tensors of a tensor operator are also computed in terms of the roots appearing in the tensor identities.

Type
Research Article
Information
The ANZIAM Journal , Volume 20 , Issue 3 , June 1978 , pp. 290 - 314
Copyright
Copyright © Australian Mathematical Society 1978

References

[1]Baird, G. E. and Biedenharn, L. C., J. Math. Phys. 5 (1964), 1730–1747.CrossRefGoogle Scholar
[2]Bracken, A. J. and Green, H. S., J. Math. Phys. 12 (1971), 2099–2106.CrossRefGoogle Scholar
[3]Carey, A. L., Cant, A. and O'Brien, D. M., Ann. Inst. Henri Poincaré, 26A (1977), 405–429.Google Scholar
[4]Edwards, S. A., “A new approach to the eigenvalues of the Gel'fand invariants for the unitary, orthogonal and symplectic groups”, Univ. of Adelaide: Dept. Math. Phys. Preprint.Google Scholar
[5]Gould, M. D., “Applications of the characteristic identity for GL(n)”, Univ. of Adelaide: Dept. Math. Phys. Preprint.Google Scholar
[6]Green, H. S., J. Math. Phys. 12 (1971), 2106–2113.CrossRefGoogle Scholar
[7]Green, H. S., J. Austral. Math. Soc. B 19 (1975), 129–139.CrossRefGoogle Scholar
[8]Green, H. S., Hurst, C. A. and Ilamed, Y., J. Math. Phys. 17 (1976), 1367–1382.CrossRefGoogle Scholar
[9]Hannabuss, K. C., “Characteristic equations for semi-simple Lie groups”, Math. Inst. Oxford Preprint (1972).Google Scholar
[10]Humphreys, J. E., Introduction to Lie algebras and representation theory (Springer-Verlag, New York-Heidelberg-Berlin, 1972).CrossRefGoogle Scholar
[11]Kostant, B., J. Funct. Anal. 20 (1975), 257–285.CrossRefGoogle Scholar
[12]Kostant, B., Trans. Amer. Math. 93 (1975), 528.Google Scholar
[13]Nwachuku, C. O. and Rashid, M. A., ICTP Report IC/75/144 (1975).Google Scholar
[14]Okubo, S., J. Math. Phys. 16 (1975), 528.CrossRefGoogle Scholar
[15]Okubo, S., Rochester Report UP–608 (1977).Google Scholar
[16] See, for example, Humphreys, [10, p. 121].Google Scholar