A New Division Algebra Representation of $E_6$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909162655449088 |
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| author | Dray, Tevian Manogue, Corinne A. Wilson, Robert A. |
| author_facet | Dray, Tevian Manogue, Corinne A. Wilson, Robert A. |
| contents | We construct the well-known decomposition of the Lie algebra $\mathfrak{e}_8$ into representations of $\mathfrak{e}_6\oplus\mathfrak{su}(3)$ using explicit matrix representations over pairs of division algebras. The minimal representation of $\mathfrak{e}_6$, namely the Albert algebra, is thus realized explicitly within $\mathfrak{e}_8$, with the action given by the matrix commutator in $\mathfrak{e}_8$, and with a natural parameterization using division algebras. Each resulting copy of the Albert algebra consists of anti-Hermitian matrices in $\mathfrak{e}_8$, labeled by imaginary (split) octonions. Our formalism naturally extends from the Lie algebra to the Lie group $E_6\subset E_8$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_00078 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A New Division Algebra Representation of $E_6$ Dray, Tevian Manogue, Corinne A. Wilson, Robert A. Group Theory High Energy Physics - Theory Mathematical Physics We construct the well-known decomposition of the Lie algebra $\mathfrak{e}_8$ into representations of $\mathfrak{e}_6\oplus\mathfrak{su}(3)$ using explicit matrix representations over pairs of division algebras. The minimal representation of $\mathfrak{e}_6$, namely the Albert algebra, is thus realized explicitly within $\mathfrak{e}_8$, with the action given by the matrix commutator in $\mathfrak{e}_8$, and with a natural parameterization using division algebras. Each resulting copy of the Albert algebra consists of anti-Hermitian matrices in $\mathfrak{e}_8$, labeled by imaginary (split) octonions. Our formalism naturally extends from the Lie algebra to the Lie group $E_6\subset E_8$. |
| title | A New Division Algebra Representation of $E_6$ |
| topic | Group Theory High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2309.00078 |