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Results tagged with lie-algebras
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user 142627
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.
1
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Connectedness of stabilizer of regular element
Let $\mathfrak{g}$ be a complex simple Lie algebra and $x \in \mathfrak{g}$ be a regular element, i.e. its centralizer is of minimal dimension.
Consider the adjoint action of the adjoint group $G$ (w …
- 617
3
votes
Regular nilpotent element in complex simple Lie algebra
There are two notions, unfortunately both called regular, which are completely different:
First there is the notion of a regular element in a Lie algebra $\mathfrak{g}$ (see for instance Serre's book …
- 617
1
vote
1
answer
570
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Lie algebra elements commuting with a principal nilpotent element
Let $\mathfrak{g}$ be a semisimple complex Lie algebra, $A \in \mathfrak{g}$ a principal nilpotent element (i.e. its centralizer is of dimension equal to the rank of $\mathfrak{g}$). I wish to underst …
- 617
2
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0
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Commutator space of regular nilpotent elements
This is a follow-up to this question.
Let $\mathfrak{g}$ be a semisimple complex Lie algebra, $A \in \mathfrak{g}$ a regular nilpotent element (i.e. its centralizer is of dimension equal to the rank …
- 617
6
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2
answers
801
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Cyclic vectors in irreducible representations of simple Lie algebras
Is there a notion of "cyclic element" in a simple Lie algebra? In particular, is it independent of the irreducible representation chosen?
Explanation.
An endomorphism A is called cyclic if there …
- 617
4
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2
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Double centralizer in special linear algebra
It is well known that for a matrix $A$ in $\mathfrak{sl}_n(\mathbb{C})$, we have the following equivalence:
$$\dim Z(A) \text{ is minimal} \leftrightarrow A \text{ is cyclic}$$
where $Z(A)$ is the cen …
- 617
0
votes
Accepted
Double centralizer in special linear algebra
I just found a counter-example for $\mathfrak{sl}_3$.
Take $A= \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ and
$B = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bm …
- 617