Special pure gradings on simple Lie algebras of types $E_6$, $E_7$, $E_8$
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| Format: | Preprint |
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2025
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| _version_ | 1866912962642444288 |
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| author | Draper, Cristina Elduque, Alberto Kochetov, Mikhail |
| author_facet | Draper, Cristina Elduque, Alberto Kochetov, Mikhail |
| contents | A group grading on a semisimple Lie algebra over an algebraically closed field of characteristic zero is special if its identity component is zero; it is pure if at least one of its components, other than the identity component, contains a Cartan subalgebra. We classify special pure gradings on Lie algebras of types $E_6$, $E_7$, $E_8$ up to equivalence and up to isomorphism. To this end, we use quadratic forms over the field of two elements to show that there are exactly three equivalence classes for $E_6$, four for $E_7$, and five for $E_8$. The computation of the corresponding Weyl groups and their actions on the universal groups yields a set of invariants that allow us to distinguish the isomorphism classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03762 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Special pure gradings on simple Lie algebras of types $E_6$, $E_7$, $E_8$ Draper, Cristina Elduque, Alberto Kochetov, Mikhail Rings and Algebras 17B70 A group grading on a semisimple Lie algebra over an algebraically closed field of characteristic zero is special if its identity component is zero; it is pure if at least one of its components, other than the identity component, contains a Cartan subalgebra. We classify special pure gradings on Lie algebras of types $E_6$, $E_7$, $E_8$ up to equivalence and up to isomorphism. To this end, we use quadratic forms over the field of two elements to show that there are exactly three equivalence classes for $E_6$, four for $E_7$, and five for $E_8$. The computation of the corresponding Weyl groups and their actions on the universal groups yields a set of invariants that allow us to distinguish the isomorphism classes. |
| title | Special pure gradings on simple Lie algebras of types $E_6$, $E_7$, $E_8$ |
| topic | Rings and Algebras 17B70 |
| url | https://arxiv.org/abs/2507.03762 |