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

Cristina Draper, Alberto Elduque, Mikhail Kochetov

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.

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

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 , , 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 , four for , and five for . 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.

Paper Structure

This paper contains 9 sections, 17 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadratic forms in characteristic 2
  4. Weyl groups of Lie algebras of the E types
  5. Gradings on Lie algebras
  6. Weyl groups of special pure gradings
  7. Special pure gradings on simple Lie algebras of types E6, E7, or E8
  8. Weyl groups of the special pure gradings on simple Lie algebras of types E6, E7, or E8
  9. Classification of special pure gradings up to isomorphism

Key Result

Proposition 2.1

Let $\mathcal{L}$ be a simple Lie algebra of type $E_6$, $E_7$, or $E_8$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$. Then the image of the homomorphism $\rho$ is the orthogonal group $\mathrm{O}(q):=\mathrm{O}(\bar{R},q)$, and the kernel is $\{\pm I\}$. Thus, the Weyl grou

Theorems & Definitions (46)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6: PaZa
  • Example 2.7
  • Definition 2.8: PaZa
  • Example 2.9
  • ...and 36 more