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# Questions tagged [lie-algebras]

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. See also the [Wiki page](http://en.wikipedia.org/wiki/Lie_algebra).

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### De Rham cohomology of homogeneous spaces

Take a homogeneous space $K/L$, where $K$ and $L$ are compact lie groups. Denoting by $\Omega^*(K/L)$ its de Rham complex, which is a homogeneous vector bundle over $K/L$, and hence has a ...
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### What subalgebras of $\mathfrak{so}(2n)$ or $\mathfrak{sp}(2n)$ are orthogonal to the centre of $\mathfrak{gl}(n)$?

Consider the Lie algebra inclusions $\mathfrak{gl}(n) \subset \mathfrak{so}(2n)$ and $\mathfrak{gl}(n) \subset \mathfrak{sp}(2n)$. Let $\mathfrak{c} \subset \mathfrak{gl}(n)$ denote the centre. ...
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### Branching rules for E6 into SU(3)^3

I am very confused about what are the branching rules for representations of $E6$ into a $SU(3)\times SU(3)\times SU(3)$ subgroup. At least in the physics literature, there seems to be a serious ...
179 views

### $L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles

First of all I want to apologize for the much too long post. A Lie group $G$ is acting on a smooth manifold $M$, then we define \begin{align*} T^k_G(M)= (S^\bullet \mathfrak{g}\otimes T^k_\mathrm{...
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### Representation theory [closed]

Let $L$ be a complex semisimple Lie algebra, then by Weyl's theorem every finite-dimensional representation of $L$ is completely reducible. My question is: can we find an example showing that in the ...
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### Finite dimensional Lie algebras: bracket of generalized eigenspaces of a derivation

Let $\delta$ be a derivation of a complex Lie algebra L, and for $\lambda \in C$, let $$L_{\lambda}=\lbrace x \in L:(\delta-\lambda 1_{L})^{m}\;x=0 \mbox{ for some } m \ge 1 \rbrace$$ be the ...
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### On maximal closed connected subgroups of a compact connected semisimple Lie group?

Let $G$ be a compact connected semisimple Lie group and let $\mathfrak g$ denote its Lie algebra. Is the following result true? Does it follows directly from Dynkin's classification of maximal Lie ...
341 views

### Weight spaces of representations of finite dimensional simple Lie algebras

This question has probably been asked before on this website, but I could not find any solution and neither can I solve this question. So again I am asking the following question: Let $\mathfrak{g}$ ...
116 views

### Number of real roots in type $\tilde{E}_8$

Let $\Phi_+$ be the set of all positive roots for a Kac-Moody algebra. Denote by $\alpha_i$ the simple root associated with node $i$ by for $i \in \{1, \ldots, n-1\}$ and by $\beta$ the simple root ...
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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 ...
102 views

### 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 ...
101 views

### Lie algebras with unique invariant bilinear symmetric form

A complex, finite dimensional Lie algebra admitting a (up to multiplication by a non-zero scalar) unique, invariant, symmetric, non-degenerate bilinear form is 1-dimensional or simple. Cf 'Lie ...

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