The split $5$-Casimir operator and the structure of $\wedge \mathfrak{ad}^{\otimes 5}$
- Autores: Isaev A.P.1,2, Krivonos S.O.1,3
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Afiliações:
- Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna, Moscow Region
- Lomonosov Moscow State University, Faculty of Physics
- Tomsk State University of Control Systems and Radioelectronics
- Edição: Volume 89, Nº 1 (2025)
- Páginas: 18-29
- Seção: Articles
- URL: https://ogarev-online.ru/1607-0046/article/view/303935
- DOI: https://doi.org/10.4213/im9594
- ID: 303935
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Resumo
In the present paper, using the split Casimir operators,we find the decomposition of the antisymmetric part ofthe fifth power of the adjoint representation$\mathfrak{ad}^{\otimes 5}$. This decomposition contains, in addition to the representations that appearesin the decomposition of $\mathfrak{ad}^{\otimes 4}$, only one newrepresentation of $X_5$. The universal dimension of this representationfor exceptional Lie algebras was proposed in [1].Our decomposition holds for all Lie algebras.
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Sobre autores
Aleksei Isaev
Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna, Moscow Region; Lomonosov Moscow State University, Faculty of Physics
Email: isaevap@theor.jinr.ru
Doctor of physico-mathematical sciences, Professor
Sergei Krivonos
Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna, Moscow Region; Tomsk State University of Control Systems and Radioelectronics
Autor responsável pela correspondência
Email: isaevap@theor.jinr.ru
Bibliografia
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- J. M. Landsberg, L. Manivel, “A universal dimension formula for complex simple Lie algebras”, Adv. Math., 201:2 (2006), 379–407
- M. Avetisyan, A. P. Isaev, S. O. Krivonos, R. Mkrtchyan, “The uniform structure of $mathfrak g^{otimes 4}$”, Russ. J. Math. Phys., 31:3 (2024), 379–388
- P. Deligne, “La serie exceptionnelle de groupes de Lie”, C. R. Acad. Sci. Paris Ser. I Math., 322:4 (1996), 321–326
- P. Vogel, The universal Lie algebra, Ecole d'ete Grenoble, 1999, 20 pp.
- А. П. Исаев, А. А. Проворов, “Проекторы на инвариантные подпространства представлений $operatorname{ad}^{otimes 2}$ алгебр Ли $so(N)$ и $sp(2r)$ и параметризация Вожеля”, ТМФ, 206:1 (2021), 3–22
- A. P. Isaev, S. O. Krivonos, “Split Casimir operator for simple Lie algebras, solutions of Yang–Baxter equations, and Vogel parameters”, J. Math. Phys., 62:8 (2021), 083503, 33 pp.
- A. P. Isaev, S. O. Krivonos, A. A. Provorov, “Split Casimir operator for simple Lie algebras in the cube of $mathrm{ad}$-representation and Vogel parameters”, Internat. J. Modern Phys. A, 38:6-7 (2023), 2350037, 29 pp.
- A. P. Isaev, V. A. Rubakov, Theory of groups and symmetries. Finite groups, Lie groups, and Lie algebras, World Sci. Publ., Hackensack, NJ, 2018, xv+458 pp.
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