Questions tagged [modular-lie-algebras]
Lie algebras in positive characteristic (not necessarily restricted Lie algebras)
27
questions
9
votes
0
answers
345
views
How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?
$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
7
votes
1
answer
293
views
Do you know a survey of modular Lie algebras and its representations?
When I was an university student, I liked reading some books about the representation theory of finite groups or Lie algebras and I was interested in explicit constructions of irreducible ...
2
votes
1
answer
104
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 ...
2
votes
0
answers
44
views
Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
4
votes
0
answers
127
views
7D simple Lie algebras over $\mathbb{F}_3$
Up to isomorphism, what are all the seven-dimensional simple Lie algebras over the field with three elements?
2
votes
0
answers
132
views
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 ...
3
votes
1
answer
120
views
Schur multiplier of finite-dimensional simple Lie algebras in positive characteristic
The Schur multipliers of finite simple groups are known and easily accessible:
https://en.wikipedia.org/wiki/List_of_finite_simple_groups
Moreover, as a consequence of the second Whitehead's Lemma, if ...
3
votes
1
answer
245
views
Jordan decomposition on the dual Lie algebra
$\newcommand\fg{\mathfrak g}\newcommand\gl{\mathfrak{gl}}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field, and let $G$ be a smooth, affine algebraic ...
5
votes
0
answers
163
views
Finite simple groups of automorphisms of finite simple Lie algebras
I begin by briefly recalling some basic facts in order to pose my question in context.
According to the classification, the finite simple groups are cyclic of prime order, are alternating on $n \geq 5$...
3
votes
1
answer
425
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 ...
15
votes
3
answers
3k
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 $\...
6
votes
1
answer
211
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.
...
7
votes
2
answers
954
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 ...
0
votes
1
answer
96
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 ...
9
votes
2
answers
796
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 ...
7
votes
2
answers
782
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
$p$...
3
votes
1
answer
199
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.
...
11
votes
0
answers
663
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 ...
3
votes
1
answer
464
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(...
0
votes
0
answers
81
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)...
5
votes
2
answers
969
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
2
answers
272
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 $...
1
vote
1
answer
240
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 ...
8
votes
3
answers
1k
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 ...
2
votes
1
answer
750
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}...
10
votes
1
answer
327
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,...
12
votes
3
answers
2k
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 $n$...