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Results tagged with lie-algebras
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user 24386
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
3
votes
Characterisation of parabolic subalgebras: reference sought
If $\mathfrak{p}^\perp$ is a nipotent subalgebra then it must consist of nilpotent elements. Indeed, let $h\in \mathfrak{p}^\perp$ and suppose ${\rm ad}\,h$ is not nilpotent.
Let $\mathfrak{g}^0(h)$ b …
- 2,624
2
votes
Borel subgroups of centralisers of Lie algebra elements in bad characteristic
I think the positive answer to this question follows from some results obtained in the paper The Hesselink stratification of nullcones and base change, Invent. Math., 191 (2013), 631-669, by M. Clarke …
- 2,624
6
votes
Accepted
Integral lattices in Lie group representations
Let $\mathfrak{n}_-$ be the Lie algebra of the derived subgroup $N_-$ of $B_{-}$ where $B_{-}=TN_-$ is a Borel subgroup of $G$ opposite to a Borel subgroup $B_+\subset P$ (here $T$ is a split maximal …
- 2,624
3
votes
Accepted
Centralizers in Jacobson-Witt Lie algebras
For $W(n,1)={\rm Der}(\mathcal{O}_n)$, defined over an algebraically filed $k$ of characteristic $p>2$, the smallest dimension of centralizers equals $n$ (here $\mathcal{O}_n$ is the $k$-algebra $k[X_ …
- 2,624
7
votes
Lie algebra of a p-group
I'm not an expert but it looks like some interesting simple Lie algebras do appear this way, yet one does not know how many. If the base field $k$ is $\mathbb{F}_p$, where $p$ is odd, and $G=\mathbb{Z …
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7
votes
Can one show the equivalence of the abstract and classical Jordan decompositions for simple ...
This is how I do this in my third year course on Lie algebras: Since we may assume that the Killing form $\kappa$ of $\mathfrak g$ is is non-degenerate, we can make use of the direct sum decomposition …
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4
votes
Difference of adjacent dominant weights is a root?
A more precise description of positive roots $\gamma$ such that $\mu=\lambda-\gamma$ can be found in the paper
http://iopscience.iop.org/article/10.1070/SM1988v061n01ABEH003200/meta
See, in partic …
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10
votes
Semisimplicity of Lie algebra in positive characteristic
If $g\subset M_n(F)$ and $n\le p-2$ then $g$ is semisimple if and only if the Killing form of g is non-degenerate. This statement is clear (and empty) if $p\in\{2,3\}$, while for $p>3$ one can use var …
- 2,624
5
votes
A strong relationship between $\mathrm{ad}(X)$ and $1-\mathrm{Ad}_g$ when $\mathrm{Ad}_gX=X$
Your conjecture that ${\rm im} (1-{\rm Ad}_g)\subseteq {\rm im}( {\rm ad}\,X)$ is indeed true. Let me sketch the argument.
First we reduce to the case where the regular element $X\in \mathfrak{g}$ i …
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4
votes
Accepted
Where does the algebraic closure enter into Block's Theorem?
Block's theorem does not require the base field $k$ to be algebraically closed but one has to be careful when $k$ is imperfect. Then $k$ will admit field extensions of the form $K=k(a)$ with $a\not\in …
- 2,624
8
votes
Accepted
Jacobson-Morozov theorem
To each unipotent element $u\in G$ one assigns its weighted Dynkin diagram which is basically
a map $\Delta\colon\, \Pi\rightarrow \{0,1,2\}$ where $\Pi$ is a basis of simple roots of the root syste …
- 2,624
16
votes
Accepted
Simple Lie algebras and Jordan decomposition
Any finite dimensional simple Lie algebra over an algebraically closed field of characteristic $p>3$
contains a nonzero element $x$ such that $ad(x)$ is semisimple. This is a nontrivial fact, and the …
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1
vote
Irreducible quotient of $U\otimes V$
Your notation suggests that that $u^-$ is a lowest weight vector, so I will asssume that this is the case. Then $u^-\otimes v^+$ generates $U\otimes V$. Indeed, let $W$ be the $\mathfrak{sl}_n$-submod …
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12
votes
finite dimensional irreducible representation of finite dimensional nilpotent Lie algebra
This is VERY far from being true: consider the $3$-dimensional Heisenberg Lie algebra
$L$ with basis $a$, $b$, $c$ and the only nonzero bracket $[a,b]=c$ (so $c$ is central in $L$). Consider the linea …
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4
votes
Accepted
Good even grading and principal Levi type
If $e$ is principal in a proper Levi subalgebra whose Dynkin diagram involves a component of type
$A_k$ with $k$ odd, then e is not even. This is very easy to see by writing down an explicit $sl_2$-tr …
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