# 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).
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### 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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### Simple restricted but not restricted simple Lie algebras

Let $F$ be a field which has a positive characteristic $p \ge 2$ and $(\mathfrak{g},[p])$ be a restricted Lie algebras over a field $F$ where $[p]$ is a $p$-th power map on $\mathfrak{g}$. $(\mathfrak{...

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### Integral Milnor-Moore theorem

Given a field K of char. zero the theorem of Milnor Moore
states that taking the enveloping hopf algebra defines an embedding
$\mathcal{U} $ from Lie algebras over K into hopf algebras over K.
Taking ...

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### Are there torsion-free restricted simple Lie algebras?

It is known that a torsion-free group can be simple (see e.g. Rataggi's paper https://www.degruyter.com/abstract/j/jgth.2007.10.issue-3/jgt.2007.028/jgt.2007.028.xml). I would like to know if the ...

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### A condition on minimal restricted subalgebras of a restricted Lie algebra

Let $L$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. Assume that the following condition holds:
For every restricted ideal $I$ of $L$, the minimal restricted subalgebras ...

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### Centralizers in Jacobson-Witt Lie algebras

Recall the (Jacobson-)Witt Lie algebras in positive characteristic: $W(n,1)$ is the Lie algebra of derivations of $\Bbbk[X_1,\dots,X_n]/(X_1^p,\dots,X_n^p)$. (For simplicity; more generally, I'm ...

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### Restricted Lie algebras with no nonzero proper restricted subalgebras

Let $L\neq 0$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. If $F$ is algebraically closed, then it is known that $L$ has no nontrivial restricted subalgebras if and only ...

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### Restricted universal extensions and lifting of derivations

Let $L$ be a perfect Lie algebra. Then it is well-known that $L$ has a universal central extension $\hat{L}$ and every derivation of $L$ can be lifted to a derivation of $\hat{L}$. (See e.g. Section 2 ...

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### Generic properties of families of algebras over an infinite dimensional base space

Let $\mathbb{k}$ be an algebraically closed field and let $A$ be a $\mathbb{k}$-algebra which is a free module of rank $r$ over some central subalgebra $Z_0$. If $Z_0$ is affine and $r$ is a finite ...

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### Restricted Lie algebras with a $p$-nilpotent basis

Let $L$ be a finite-dimensional restricted Lie algebra over a field of characteristic $p>0$. An element $x$ of $L$ is called $p$-nilpotent if $x^{[p]^k}=0$ for some positive integer $k$. If $L$ is ...

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### Vanishing of power of nilpotent operator $\mathrm{ad} \, \;e$ in different characteritics

This question needs some background:
(1) In his influential 1959 paper here, Kostant studied the adjoint representation of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (which can be ...

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### indecomposable modules restricted from $gl_n$ to $sl_n$

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra.
Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced ...

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### Computation of restricted Lie algebra (co)homology

My question is the following:
Is there a small complex, perhaps analogous to the Chevalley-Eilenberg complex, computing the (co)homology of a restricted Lie algebra over a field of characteristic $...

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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). ...

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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 ...

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### 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 ...

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### Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?

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 ...

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### Restricted universal enveloping algebra of Abelian p-Lie algebra

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 $k$-...