Constructing Chayet-Garibaldi algebras from affine vertex algebras (including the 3876-dimensional algebra for $E_8$)
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909989614911488 |
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| author | De Medts, Tom Olyslager, Louis |
| author_facet | De Medts, Tom Olyslager, Louis |
| contents | In 2021, Maurice Chayet and Skip Garibaldi provided an explicit construction of a commutative non-associative algebra on the second smallest representation of $E_8$ (of dimension $3875$) adjoined with a unit. In fact, they define such an algebra $A(\mathfrak{g})$ for each simple Lie algebra $\mathfrak{g}$, in terms of explicit but ad-hoc formulas.
We discovered that their algebras $A(\mathfrak{g})$ have a natural interpretation in terms of affine vertex algebras, and their ad-hoc formulas take an extremely simple form in this new interpretation. It is our hope that this point of view will lead to a better understanding of this interesting class of algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_24103 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructing Chayet-Garibaldi algebras from affine vertex algebras (including the 3876-dimensional algebra for $E_8$) De Medts, Tom Olyslager, Louis Rings and Algebras Group Theory Representation Theory 20G41, 17B45, 17B69, 20G05, 20G15 In 2021, Maurice Chayet and Skip Garibaldi provided an explicit construction of a commutative non-associative algebra on the second smallest representation of $E_8$ (of dimension $3875$) adjoined with a unit. In fact, they define such an algebra $A(\mathfrak{g})$ for each simple Lie algebra $\mathfrak{g}$, in terms of explicit but ad-hoc formulas. We discovered that their algebras $A(\mathfrak{g})$ have a natural interpretation in terms of affine vertex algebras, and their ad-hoc formulas take an extremely simple form in this new interpretation. It is our hope that this point of view will lead to a better understanding of this interesting class of algebras. |
| title | Constructing Chayet-Garibaldi algebras from affine vertex algebras (including the 3876-dimensional algebra for $E_8$) |
| topic | Rings and Algebras Group Theory Representation Theory 20G41, 17B45, 17B69, 20G05, 20G15 |
| url | https://arxiv.org/abs/2503.24103 |