Vogel universality and beyond
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912868055646208 |
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| author | Isaev, A. P. |
| author_facet | Isaev, A. P. |
| contents | For simple Lie algebras we construct characteristic identities for split (polarized) Casimir operators in representations $T \otimes Y_n$ and $T \otimes Y_n'$, where $T$ -- defining (minimal fundamental for exceptional Lie algebras) representation, $Y_n$ -- n-Cartan powers of the adjoint representations $ad = Y_1$ and Y_n' -- special representations appeared in the Clebsch-Gordan decomposition of symmetric part of $ad^{\otimes n}$. By means of these characteristic identities, we derive (for all simple Lie algebras, except $\mathfrak{e}_8$) explicit formulae for invariant projectors onto irreducible subrepresentations arose in the decomposition of $T \otimes Y_n$. These projectors and characteristic identities are written in the universal form for all simple Lie algebras (except $\mathfrak{e}_8$) in terms of Vogel parameters. Universal formulas for the dimensions of the Casimir subrepresentations appeared in the decompositions of $T \otimes Y_n$ where found. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01612 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vogel universality and beyond Isaev, A. P. Mathematical Physics High Energy Physics - Theory 81R05, 81R12, 81T13, 81T18, 81V22, 17B20, 17B25, 33C52 For simple Lie algebras we construct characteristic identities for split (polarized) Casimir operators in representations $T \otimes Y_n$ and $T \otimes Y_n'$, where $T$ -- defining (minimal fundamental for exceptional Lie algebras) representation, $Y_n$ -- n-Cartan powers of the adjoint representations $ad = Y_1$ and Y_n' -- special representations appeared in the Clebsch-Gordan decomposition of symmetric part of $ad^{\otimes n}$. By means of these characteristic identities, we derive (for all simple Lie algebras, except $\mathfrak{e}_8$) explicit formulae for invariant projectors onto irreducible subrepresentations arose in the decomposition of $T \otimes Y_n$. These projectors and characteristic identities are written in the universal form for all simple Lie algebras (except $\mathfrak{e}_8$) in terms of Vogel parameters. Universal formulas for the dimensions of the Casimir subrepresentations appeared in the decompositions of $T \otimes Y_n$ where found. |
| title | Vogel universality and beyond |
| topic | Mathematical Physics High Energy Physics - Theory 81R05, 81R12, 81T13, 81T18, 81V22, 17B20, 17B25, 33C52 |
| url | https://arxiv.org/abs/2601.01612 |