Questions tagged [modular-lie-algebras]

Lie algebras in positive characteristic (not necessarily restricted Lie algebras)

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14
votes
3answers
2k views

Which is the correct universal enveloping algebra in positive characteristic?

This is an extension of this question about symmetric algebras in positive characteristic. The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers. Let $\...
11
votes
0answers
406 views

Is there a reasonable way to define “reductive Lie algebra” in prime characteristic?

Among the finite dimensional Lie algebras over a field of characteristic 0, there is a sensible definition of "reductive Lie algebra" going back at least to the 1960 first chapter of N. Bourbaki's ...
10
votes
3answers
1k views

Semisimplicity of Lie algebra in positive characteristic

Let $F$ be a field of characteristic $p > 0$. Let $\mathfrak{g}$ be a linear Lie algebra, that is $\mathfrak{g}\subset M_n(F)$ for some natural number $n$. Does there exist a condition involving $n$...
10
votes
1answer
268 views

An identity in Lie algebras over fields of positive characteristic

Let $L$ be a Lie algebra over a field of characteristic $p>0$ and $D$ a derivation of $L$. For every $x\in L$ denote by $\mathrm{ad} x$ the adjoint map $\mathrm{ad}x: L \rightarrow L, a\mapsto [x,...
9
votes
2answers
625 views

On nilpotency of the derived subalgebra of a solvable Lie algebra

Using Engel's Theorem and Lie's Theorem, one can easily establish the following result: Let $ \frak{g} $ be a finite-dimensional Lie algebra over an algebraically closed field $ \mathbb{F} $ of ...
8
votes
3answers
958 views

Failure of Jacobson-Morozov in positive characteristics

The Jacobson-Morozov theorem that any nilpotent element $e$ in the Lie algebra of a simple algebraic group $G$ can be embedded in an $\mathfrak{sl}_2$-triple, has a restriction (in terms of the ...
7
votes
2answers
794 views

Kostant's theorem on invariant polynomials in positive characteristic

Let $k$ be an algebraically closed field and let $G$ be a reductive linear algebraic group over $k$ with Lie algebra $\mathfrak g$. If the characteristic of $k$ is $0$ then, by a classical result of ...
6
votes
2answers
659 views

Lie algebras and non-smoothness of centralisers in bad characteristic

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p>0$. For $x\in G$, let $C_{G}(x)$ denote the centraliser, considered as a group scheme over $k$. If $p$...
6
votes
1answer
124 views

Maximal dimension of abelian subalgebra of exceptional simple Lie algebra in positive characteristic

For complex semisimple Lie algebras, the maximal dimension of an abelian subalgebra was determined by Mal'cev in 1945. For $E_7$, for example, it is $27$, and is the radical of the $E_6$ parabolic. ...
4
votes
2answers
605 views

Lie's theorem in characteristic $p$

Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...
3
votes
1answer
367 views

Harish-Chandra isomorphism for characteristic $p$

I am trying to understand the proof of Theorem 1 from this paper V. Kac and B. Weisfeiler (Indag. Math. 1976, DOI link). Theorem 1. Let either $p\neq 2$ or $\varrho\in X(\mathscr{T})$. Then $\gamma(...
3
votes
1answer
159 views

Borel subgroups of centralisers of Lie algebra elements in bad characteristic

Let $G$ be a simple linear algebraic group over an algebraically closed field $k$ of characteristic $p>0$, and let $\mathfrak{g}=\mathrm{Lie}(G)(k)$ denote (the $k$-points of) the Lie algebra. ...
3
votes
2answers
236 views

Replacement for Lie-algebra complements

All groups are linear algebraic over some fixed field $k$. I believe that it is true that, in characteristic $0$, if $G'$ is a reductive subgroup of $G$, then there is a $G'$-invariant complement to $...
2
votes
1answer
659 views

Branching rule for classical Lie algebras in positive characteristic

The restriction of an irreducible $\mathfrak{sl}_n(\mathbb{C})$-module to $\mathfrak{sl}_{n-1}(\mathbb{C})$ is described by a branching rule which says that if $L(\lambda)$ is the simple $\mathfrak{sl}...
2
votes
1answer
293 views

Solvable Lie algebra whose nilradical is not characteristic

Say that an ideal in a Lie algebra is characteristic if it is invariant under every derivation of the algebra. It is well known that the nilradical of a finite-dimensional Lie algebra over a field ...
1
vote
1answer
155 views

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 ...
0
votes
1answer
76 views

Automorphisms of Lie algebra of type $A_5$ modulo its center in characteristic 2

Let $L$ be classical Lie algebra of type $A_5$ over field of characteristic 2; let $M$ be the quotient $L/Z(L)$ modulo its center $Z(L)$. What about the group of automorphisms of M? Does anybody ...
0
votes
0answers
68 views

Format of grading Witt Lie Algebra

Let $W(n,m)$ be generalized Jacobson-Witt algebra over a field of characteristic $p>3$. According to the grading of $W(n,m)$, we know that it inherits the grading from $A(n,m)$ as follows: $$W(n,m)...