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arXiv · 2507.03762

Special pure gradings on simple Lie algebras of types $E_6$, $E_7$, $E_8$

Cristina Draper·Alberto Elduque·Mikhail Kochetov
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Abstract

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.

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Cristina Draper, Alberto Elduque, Mikhail Kochetov. 2025-07-04. Special pure gradings on simple Lie algebras of types $E_6$, $E_7$, $E_8$. https://doi.org/10.1016/j.laa.2026.02.019

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