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Results tagged with lie-algebras
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user 24965
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.
16
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5
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About the intrinsic definition of the Weyl group of complex semisimple Lie algebras
It may be a easy question for experts.
The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h} …
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10
votes
3
answers
632
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Is the tensor product of two infinite dimensional objects in the BGG category $\mathcal{O}$ ...
Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra (according to a comment of Victor Ostrik, we need to further require that $\mathfrak{g}$ is simple) and we can consider its BG …
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8
votes
3
answers
694
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What is the categorical significance of the trivial $\mathfrak{g}$-module in the category of...
This question may be trivial for experts.
Let $\mathfrak{g}$ be a Lie algebra over a field $k$ and consider the category $\mathfrak{g}$-mod of $\mathfrak{g}$ modules. We can add suitable conditions, …
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7
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2
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417
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About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$
Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) …
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7
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0
answers
166
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How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in th...
Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be t …
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6
votes
0
answers
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The meaning of a "subcomplex" of the Cartan-Eilenberg of a Lie algebra
Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^* $ be the dual vector space. We have the standard Cartan-Eilenberg complex
$(\wedge^{\cdot} \mathfrak{g}^* ,\text{d}_{C …
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6
votes
1
answer
400
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Does the vanishing of the Poisson bracket on $S(\mathfrak{g})^{\mathfrak{g}}$ inspire the di...
For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. In fact let's define …
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5
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1
answer
419
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Does $\text{Mat}_n(k)$ have some universal properties similar to its universal enveloping al...
Let $k$ be a field and $\text{Mat}_n(k)$ be $n \times n$ matrices over $k$. Let's consider $\text{Mat}_n(k)$ as an associative algebra and denote $gl_n(k)$ be the same $k$-linear space as $\text{Mat}_ …
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5
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2
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915
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Could we define the semi-direct product of two universal enveloping algebras?
If we have two Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ over a field $k$, and if we have a Lie algebra homomorphism $\mathfrak{g}\rightarrow \text{Der}_k(\mathfrak{h})$, then we can define the s …
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4
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Is one of the hyperplane partitions of a irreducible root system always generate the whole W...
Let $\Delta$ be a irreducible root system and $\Delta^+$ be its positive roots.
We say a subset $\Delta^{\prime}\subset \Delta^+$ can generate the Weyl group if reflections of roots in $\Delta^{\prim …
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4
votes
1
answer
323
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About the term "tangential derivation" on a free Lie algebra.
Let $\mathcal{lie}_n$ be the free Lie algebra generated by $n$ elements $x_1,\ldots, x_n$. A derivation $u\in \text{Der}(\mathcal{lie}_n)$ is called tangential if there exist $a_i\in \mathcal{lie}_n, …
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4
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2
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What are the "tensor-closed" object of the BGG category $\mathcal{O}$ of a semisimple Lie al...
Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra and we can consider its BGG category $\mathcal{O}$. It is well-known that $\mathcal{O}$ is not closed under tensor product, i. …
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4
votes
1
answer
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Can we have a nontrivial division of a irreducible root system as the union of two closed su...
The question is related to this MO question. Let $(\Phi, E)$ be a irreducible crystallographic root system where $\Phi$ is the set of all roots and $E$ is the $\mathbb{R}$-span of $\Phi$. As in the st …
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3
votes
Closure relations between Bruhat cells on the flag variety
For a first introduction you can read Michel Brion's "http://arxiv.org/pdf/math/0410240v1.pdf". He gives a nice introduction (for G=GL(n)) in Section 1.
I'm not sure whether your curve method works b …
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3
votes
Can we have a nontrivial division of a irreducible root system as the union of two closed su...
$\def\abs#1{\lvert#1\rvert}\DeclareMathOperator\Span{Span}$I think I get a proof inspired by the comment of @LSpice.
First we can prove that $\Phi_1\setminus \Phi_2$ is orthogonal to $\Phi_2\setminus …
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