Hostname: page-component-669899f699-vbsjw Total loading time: 0 Render date: 2025-04-26T00:28:36.550Z Has data issue: false hasContentIssue false
  • English
  • Français

Beyond the Enveloping Algebra of sl3

Published online by Cambridge University Press:  20 November 2018

Daniel E. Flath  and
L. C. Biedenharn
Daniel E. Flath
Affiliation:
Duke University, Durham, North Carolina
L. C. Biedenharn
Affiliation:
Duke University, Durham, North Carolina
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The problem which motivated the writing of this paper is that of finding structure behind the decomposition of the sl3 representation spaces V* ⊗ W = Hom(V, W) for finite dimensional irreducible sl3-modules V and W. For sl2 this extends the classical Clebsch-Gordon problem. The question has been considered for sl3 in a computational way in [5]. In this paper we build a conceptual algebraic framework going beyond the enveloping algebra of sl3.

For each dominant integral weight α let Vα be an irreducible representation of sl3 of highest weight α. It is well known that, for weights α, μ, λ, the multiplicity of Vλ in Hom(Vα, Vα+μ) is bounded by the multiplicity of μ in Vλ, with equality for generic α. This suggests the possibility of a single construction of highest weight vectors of weight X in Hom(Vα, Vα+μ) which is valid for all a.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 4 , 01 August 1985 , pp. 710 - 729
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Baird, G. E. and Biedenharn, L. C. On the representations of the semi-simple lie groups IV. A canonical classification for tensor operators in SU3, J. Math. Phys. 5 (1964), 1730–1747.Google Scholar
2. Bourbaki, N., Groupes et algebres de lie, Chap. VI Section 4.7 (Hermann, Paris, 1968).Google Scholar
3. Gel'fand, I. M. and Zetlin, M. L., Dokl. Akad. Nauk. SSSR 71 (1950), 825–828.Google Scholar
4. Klimyk, A. U., Decomposition of a tensor product of irreducible representations of a semisimple lie algebra into a direct sum of irreducible representations, Amer. Math Soc. Translations (Amer. Math. Soc., Providence, 1968).Google Scholar
5. Louck, J. D. and Biedenharn, L. C., Canonical unit adjoint tensor operators in U(n), J. Math. Phys. 11 (1970), 2368–2414.Google Scholar
6. Weyl, H., The theory of groups and quantum mechanics (Methuen and Co., London, 1931).Google Scholar
7. Weyl, H., The structure and representation of continuous groups, Lectures at the Institute for Advanced Study (1934-1935) (notes by Richard Brauer, reprinted 1955).Google Scholar