# Questions tagged [restricted-lie-algebras]

Lie algebra over field (or ring) of prime characteristic p>0 endowed with an additional "p-operation" (see the tag 'modular-lie-algebras' for plain Lie algebras in positive characteristic). To be combined with the more generic tag 'lie-algebras'

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### Classification of restricted Lie algebras of reductive groups

$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
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### Derived subalgebra of a restricted Lie algebra

Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map). ...
101 views

### Proof of restrictableness of Lie algebra without basis

$\DeclareMathOperator\ad{ad}$According to the Strade and Farnsteiners' textbook, a Lie algebra $L$ is restrictable if and only if $\ad(L)$ is $p$-subalgebra of $\operatorname{Der}(L)$. Also, the ...
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### Automorphism groups of "reductive" Lie algebras in positive characteristic

I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras. Let $G$ be a reductive group ...
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### Classification of simple Lie algebras over finite fields

Classification of simple (or simple-restricted) Lie algebras over algebraically closed fields in positive characteristic is studied for a long time. Today, we know all finite-dimensional simple (or ...
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1 vote
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### hamilton type Lie algebras

If n be positive integer and for an n-tuple of positive integers m=(m1,...,mn) then p(n,m) is graded and filtered subalgebra of W(n,m).p(n,m) is called non-alternating hamilton lie algebra over GF(2). ...
1 vote
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### Semisimple elements in division algebras

I found the following exercise at page 85 of the Strade-Farnsteiner's book "Modular Lie algebras and their representation": Let $D$ be a finite-dimensional division ring over a field $F$ of ...
411 views

### Weights of restricted modules of some Cartan type Lie algebras

Let $L$ be a simple Lie algebra of Cartan type of absolute toral rank 2 over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq 5$. Denote by $L_{[p]}$ the minimal $p$-envelope of $L$...
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### German term for "restricted Lie algebra" ?

Can anyone tell me the German term for "restricted Lie algebra" ? Many thanks in advance ! Kind regards, Stephan Kroneck.
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### Maximal dimension of abelian ideals of a Lie algebra and extensions of the ground field

For a Lie algebra $L$ of dimension $n$ over a field ${\mathbb F}$ we denote by $\beta(L)$ the maximal dimension of abelian ideals of $L$. In general, $\beta(L)$ is not preserved under extensions of ...
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### Restricted Lie algebras of low dimension

Over the decades there has been a lot of papers devoted to the classification of Lie algebras of low dimension. Do you know any paper dealing with the problem of determining (up to restricted ...
Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$. For any $k$-vector space $V$, consider the canonical ...
Question: Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$. Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-...