# Questions tagged [semisimple-lie-algebras]

A simple Lie algebra is non-abelian Lie algebra with no nontrivial ideals. A *semisimple Lie algebra* is a Lie algebra which is the direct sum of simple Lie algebras. This tag is for questions about semisimple Lie algebras, including their classification and correspondent to root systems and Dynkin diagrams.

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### Simple modules of $\mathcal{g}\otimes\mathbb{C}[x]$ for a lie algebra $\mathcal{g}$

Let $\mathcal{g}$ be a simple, finite-dimensional complex Lie algebra, and let $\mathcal{M}$ be a representative system of finite-dimensional simple $\mathcal{g}$-modules (up to isomorphism). ...
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### Process for decomposing tensor products of Lie algebra representations into irreducibles?

I'm trying to decompose tensor products of semisimple Lie group/algebra representations into direct sums of irreducible representations. I know this question has been asked many times before, with ...
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### Other direction of Weyl's theorem on semisimple Lie-algebras

Weyl's theorem states Let $\mathfrak{g}$ be a finite dimensional semisimple Lie-algebra over an algebraically closed field with characteristic 0. Then all finite dimensional representations are ...
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### Finite subgroups of the automorphism group of sl2

My question is straightfoward: is there a classification of the finite subgroups of $\operatorname{Aut} \mathfrak{sl}_2(\mathbb{C})$? I mean of course groups of Lie algebras automorphisms. If such a ...
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### Any element of a semisimple Lie group is conjugate (via the adjoint action) to an element in the Cartan subalgebra?

The statement: If $G$ is a semisimple Lie group and $\mathfrak{g}$ the corresponding Lie algebra, then the general statement is that every element of $\mathfrak{g}$ is conjugate, via the adjoint ...
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### Problem in understanding a fact about Lie algebra.

Let $L$ be a simple complex Lie algebra. Let $\Phi$ be the root set of $L$ and $\Gamma$ be the set of all simple roots of $L.$ We know that for every root $\alpha \in \Phi$ there is a copy $S_{\alpha}$...
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### Is the tensor product of irreducible representations semisimple?

Say you have a semi-simple (in my case even simple) Lie algebra $\mathfrak{g}$ with two simple modules $V$ and $W$, which may be infinite dimensional. Is $V \otimes W$ always semi-simple? My guess ...
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### curvature tensor of symmetric spaces

I'm trying to understand the following theorem about the curvature tensor of a symmetric space: Let $R$ the curvature tensor of the space $G/K$ corresponding to the Riemannian structure $Q$ ,then at ...
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Recently, I am studying Lie algebra and have some question for the root system. Let $\mathfrak g$ be a simple Lie algebra and $\mathfrak h$ a cartan subalgebra and $\Delta$ the root system of $\... 0 votes 0 answers 101 views ### If$\alpha, \beta, \alpha + \beta \in \Phi$then$[L_{\alpha} , L_{\beta}] = L_{\alpha + \beta}.$Let$L$be a semisimple Lie algebra and$\Phi$be the root set of$L.$Let$L_{\alpha}$be the root subspaces of$L$corresponding to the root$\alpha \in \Phi.$Show that if$\alpha, \beta, \alpha + \...
I am currently reading Edward Frenkel's "Langlands Correspondence for Loop Groups", freely available here: https://math.berkeley.edu/~frenkel/loop.pdf. In the appendix $A.4$ he describes ...