Maximal subalgebras of the exceptional Lie algebras in low characteristic

Purslow, Thomas

doi:10.48550/arxiv.1803.06357

Cited by 2 publications

(3 citation statements)

References 31 publications

(120 reference statements)

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“…The algebras of type A 2n+1 , B n , C n , D n , E 7 have a non-zero center, and F 4 has a 26-dimensional ideal J generated by the short roots. We have F 4 /J ∼ = J and this is a simple Lie algebra in characteristic 2, see [4], or [29], section 5.1. Furthermore F 4 has up to isomorphism only one such simple ideal of dimension 26.…”

Section: Classical Type the First Class Of Simple Modular Lie Algebra...mentioning

confidence: 99%

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

Burde¹,

Moens²

2020

Preprint

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Commutative post-Lie algebra structures on Lie algebras, in short CPA structures, have been studied over fields of characteristic zero, in particular for real and complex numbers motivated by geometry. A perfect Lie algebra in characteristic zero only admits the trivial CPA-structure. In this article we study these structures over fields of characteristic p > 0. We show that every perfect modular Lie algebra in characteristic p > 2 having a solvable outer derivation algebra admits only the trivial CPA-structure. This involves a conjecture by Hans Zassenhaus, saying that the outer derivation algebra Out(g) of a simple modular Lie algebra g is solvable. We try to summarize the known results on the Zassenhaus conjecture and prove some new results using the classification of simple modular Lie algebras by Premet and Strade for algebraically closed fields of characteristic p > 3. As a corollary we obtain that that every central simple modular Lie algebra of characteristic p > 3 admits only the trivial CPA-structure.

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Section: Classical Type the First Class Of Simple Modular Lie Algebra...mentioning

confidence: 99%

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

Burde¹,

Moens²

2020

Preprint

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show abstract

“…One possibility is the family of simple Lie algebras si(sle(n)) of dimension 2 2n−1 − 2 n−1 − 2 for n ≥ 3, see [12], Lemma 2.2.2. This algebra is denoted by sh(2n; m) in Purslow's thesis [16], Theorem 5.4.3. We used Purslow's construction for n = 4, see [16] p. 138 − 141, to compute the outer derivation algebra of this 118-dimensional algebra.…”

Section: 3mentioning

confidence: 99%

“…This algebra is denoted by sh(2n; m) in Purslow's thesis [16], Theorem 5.4.3. We used Purslow's construction for n = 4, see [16] p. 138 − 141, to compute the outer derivation algebra of this 118-dimensional algebra.…”

Section: 3mentioning

confidence: 99%

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

Burde

Moens

2020

Journal of Algebra

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Maximal subalgebras of the exceptional Lie algebras in low characteristic

Cited by 2 publications

References 31 publications

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

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