Questions tagged [restricted-lie-algebras]

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18 questions
59 views

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 ...
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 ...