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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.
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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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When does the Kazhdan-Lusztig polynomial $P_{x,w}(q)$ not vanish at $q=1$?
Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra. For any $\lambda\in \mathfrak{h}^{*}$ let $M(\lambda)$ and $L(\lambda)$ be the Verma module and the simple mo …
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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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Can we write $sl(4,\mathbb{C})$ as the vector space sum of two copies of $sl(3,\mathbb{C})$? [closed]
We know $sl(4,\mathbb{C})$ has dimension $15$ and $sl(3,\mathbb{C})$ has dimension $8$. Is it possible to write $sl(4,\mathbb{C})$ as the vector space sum of two Lie subalgebras that are isomorphic to …
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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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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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Can we have a nontrivial division of a irreducible root system as $\Phi=\Phi_{[\lambda]}\cup...
Let $(\mathfrak{g},\mathfrak{h},\Phi)$ be a root system of a complex simple Lie algebra, where $\Phi$ is the set of all roots. For each $\alpha\in \Phi$, let $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$ be …
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Can we have a nontrivial division of a irreducible root system as $\Phi=\Phi_{[\lambda]}\cup...
I think the answer is yes because $(\Phi_{[\lambda]})^{\vee}$ and $(\Phi_{[\mu]})^{\vee}$ are closed sub-root systems of the dual root system $\Phi^{\vee}$. Closed means if $\alpha$ and $\beta$ are ro …
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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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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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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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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 …
CommunityBot
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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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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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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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