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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.
17
votes
Accepted
Failure of Jacobson-Morozov in positive characteristics
The uniqueness can break down very badly in positive characteristic. Supose $G=SL_p$ where $p$ is the characteristic of the base field. Take a regular nilpotent element $e$ in $\mathfrak{g}=\mathfrak{ …
- 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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13
votes
Accepted
Are S(g) and U(g) isomorphic as g-modules for g Lie algebra over F_p ? Are S(g)^g and U(g)^g...
Question 2 has a negative answer. Indeed, let $L =sl(2,k)$ where $k$ is an algebraically closed field of characteristic $p>3$ and let {$e,h,f$} be the standard basis of $L$. Then it is well-known (and …
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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 …
- 2,624
11
votes
About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$
Here is a complete answer to this question: the map $\varphi$ is an embedding if and only if the group $H$ contains a maximal torus of $G$. I'm assuming (as in the question) that all groups are comple …
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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 …
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10
votes
For $\mathfrak g$ A Lie algebra of type $ E_7 $, $\mathfrak h $ a Cartan subalgebra and $\De...
There is a related old paper by Tits on normalisers of tori, but my copy is long gone and I'm not sure whether the splitting issues had been addressed there. In the case of $E_7$, the sequence does no …
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9
votes
Accepted
Commutativity and Kostant sections
In some cases the answer to the weaker version of the question (involving the semisimple
part of $X_2$) is YES. This will happen if $C_g(e)$ is self-dual which is the case, for instance, when $g=gl_N$ …
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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 …
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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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7
votes
Accepted
Smallest dimension of nontrivial representation of a simple Lie algebra over `$\mathbb{C}$`
In Bourbaki, Ch. VIII, $\S$ 7, Sect. 2, one can find the notion of an $\mbox{$R$-saturated set}$, and Corollary to Prop. 4 in that section proves that for every $R$-saturated set $\mathcal X$ there is …
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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 …
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6
votes
Deformations of semisimple Lie algebras
I think that the explanation
"Because the Cartan classification of isomorphism classes of semisimples is discrete (no continuous families), connected components of the space of semisimples are alway …
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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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