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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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About root space decomposition of complex semisimple Lie algebra

It is well-known that for a complex semisimple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$ and root system $\Phi$, there is a root space decomposition $\mathfrak{g}=\mathfrak{h}\...
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Lie algebra homomorphisms preserve semi-simplicity?

Is the following proof correct? Claim: Let $g$ be a semi-simple Lie algebra, and $f: g\rightarrow h$ be a homomorphism of Lie algebras. Then $Im f \leq h$ is a semi-simple Lie algebra. Proof ...
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Is this Cartan's Criterion for Semisimplicity

I am reading through a paper on semisimplicity of adjacency algebras of association schemes and the one example uses what seems to be the Cartan Criterion for a finite dimensional field of ...
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Relation between semisimple Lie algebra completely reducibility and semisimple ring

Let $\frak g$ be a semi-simple finite dimensional Lie algebra over the complex numbers $\mathbb C$. Then every non irreducible representation of $\frak g$ is completely reducible. Q1: Is category f....
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Inner automorphisms of a real semisimple Lie algebra

There are at least two ways of defining the inner automorphisms of a real Lie algebra $\mathfrak{g}$. One is the algebraic definition: an inner automorphism is $\exp (\text{ad} X)$, where $X$ is an ...
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Proving that $G=\mathrm{SL}_d(\mathbb{R})$ is semisimple

I want to find a reference to a proof of semisimplicity of the special linear group $G=\mathrm{SL}_d(\mathbb{R})$ by showing that the Lie algebra $\mathfrak{g}$ of $G$ of matrices with trace $0$ is ...
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What is known about such dominant integral weights of compact semisimple Lie groups?

I am interested in special dominant integral weights $\lambda \in \mathfrak{h}^*$, where $\mathfrak{h}$ is a Cartan subalgebra of the Lie algebra $\mathfrak{g}$ of a compact semisimple Lie group $G$. ...
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Is this subalgebra semisimple?

Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Let $\Phi$ be the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by ...
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“If $g$ is semisimple, It is not too hard to see that $H^2(g,a)=0$. With a little supplementary argument…”

This is a statement made in Knapp, Lie groups, Lie algebras, Cohomology Chpt 4 last paragraph of Sec 2. $H^i(g,a)$ is the $i-$th cohomology group of complex $Hom(\wedge^i g,a)$ with $a$ abelian lie ...
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Question about definition related to root system of semisimple Lie algebras

Let $L$ be a semisimple Lie algebra of finite dimension over a field of charcteristic 0 and algebraically closed, and $H$ a maximal toral subalgebra. Let $R$ be the set of roots of $L$ with respect ...
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How to show this $L$-module is simple? (related to the root space decomposition of semisimle Lie algebras)

Given a semisimple Lie algebra (finite dimensional over a field $K$ characteristic $0$ and algebraically closed), there exists a root space decomposition $$ L = H \oplus \oplus_{\alpha \in R} L_{\...
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Is the semisimple part of $x \in L$ belong to $L$ when $L$ is a semisimple Lie algebra?

Let $L$ be a finite dimensional semisimple Lie algebra over a field of charcteristic $0$ and algebraically closed. I have learned that the map $$ ad: L \rightarrow \ Der(L) $$ is an isompsphism, ...
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35 views

About nondegeneration of Killing form

Let $\mathfrak{r}$ be a reductive Lie subalgebra of the complex semisimple Lie algebra $\mathfrak{g}$ and let $B$ be the Killing form of $\mathfrak{g}$. Suppose that the restriction $B|_\mathfrak{r}$ ...
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Cartan subalgebra minimal definition

One of the definitions of Cartan subalgebra $\mathfrak{h}$ of a semisimple Lie algebra $\mathfrak{g}$ one can find in the literature is that it is a maximal abelian subalgebra has the property that $...
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Classification of real semisimple lie algebras

We know that each complex semisimple lie algebra $L$ is a direct sum of a chosen Cartan subalgebra $H$ and finitely many weight spaces, each of which is associated with an element in $H^*=\...
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Finding the Weyl group from a Cartan matrix

I am looking to establish the relationship between Weyl groups and semisimple Lie algebras. So far I have found the root space decomposition (I am using $\mathfrak{sl}(3, \mathbb{C}) $as my example). ...
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Existence of a non essentially real ideal in a semisimple Lie algebra

Let $\mathfrak{g}$ be the Lie algebra of a semisimple compact Lie group $G$. Denote by $\mathbb{C} \mathfrak g = \mathbb{C} \otimes \mathfrak g$ the complexification of $\mathfrak g$. I am assuming ...
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Subgraphs of Dynkin Diagrams

Am I right in thinking that if we have two semisimple Lie Algebras $\mathfrak{g} $ and $\mathfrak{h}$ with respective Dynkin Diagrams $A$ and $B$, we may find an injective homomorphism of Lie Algebras ...
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Which roots are fixed by simple reflections of the Weyl Group?

Let $\Phi$ be a root system of a semisimple Lie Algebra, and $W$ it's Weyl group. Let $\Delta = \{ \alpha_1, \dots, \alpha_l \}$ be a root basis, and let $w_i \in W$ be the simple reflection ...
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Finite dimensional irreducible representations of a semisimple Lie Algebra separate points of the universal enveloping algebra.

Let $\mathfrak{g}$ be a semisimple Lie Algebra, and $U(\mathfrak g)$ the universal enveloping algebra . We know that for every representation $\rho: \mathfrak g \to \mathfrak{gl}(V)$, there exists a ...
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How to tell the rank of a semisimple Lie algebra?

My understanding is that the rank of a finite-dimensional semisimple Lie algebra (over an algebraically closed field of characteristic zero) is defined non-constructively as the (unique) dimension of ...
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A theorem about semisimple Lie algebra

I'm reading Lie Groups, Lie Algebras , and Representations (first edition) by Hall and I'm stuck by a theorem (The author did not prove it): Theorem 6.6 A complex Lie algebra is semisimple if and ...
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Ideals of semisimple Lie algebras

Let $\mathfrak{g} \subset gl(V)$ be a semisimple Lie algebra. I already know that symmetric bilinear form $f(x,y)=\mathbf{Trace}(XY)$ is nonsingular on $\mathfrak{g}$. And I've read that any ideal $\...
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Why is a root system called a “root” system?

Root systems plays an important role in, among other things, classifying semisimple Lie Algebras. Their name suggest that they have something to do with "roots" of a polynomial. Are they the roots of ...
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Semisimple Lie algebra representation to reductive lie algebra

Let us work over $\Bbb C$. If I know the classification of irreducible representations of semisimple Lie algebras, how do I classify the irreducible representations of reductive Lie algebras? Let $\...
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Property of semisimple Lie subalgebra

I have a question about a property of Jordan Chevalley decomposion components of any element of a semisimple Lie subalgebra $g \subset gl(V)$. If we have for any $x \in g$ a J-C decomposion $ x= x_s ...
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Cartan subalgebras of a semisimple Lie algebra [closed]

Let $L$ be a semisimple complex Lie algebra and $H$ a Cartan subalgebra. Suppose that $L=L_{1}\oplus ...\oplus L_{k}$ with each $L_{i}$ a simple ideal of $L$. Show that for $1\leq i \leq k$, $H\cap L_{...
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300 views

Semisimple algebraic group vs semisimple Lie algebra

Let $G$ be a connected affine algebraic group over $\mathbb{C}$ of positive dimension. By definition: $G$ is semisimple as an algebraic group if it has no non-trivial connected, normal, abelian ...
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If $\mathfrak{g}=\bigoplus\mathfrak{g}_i$ is a semisimple Lie algebra, why does $\mathfrak{h}=\bigoplus\left(\mathfrak{h}\cap\mathfrak{g}_i\right)$?

There is this property about Cartan subalgebras that is not clear to me. Suppose $\mathfrak{g}$ is a semisimple Lie algebra. Then I know we can decompose it uniquely as $\mathfrak{g}=\bigoplus\...
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1answer
332 views

Dimension of the root spaces of a semisimple complex Lie algebra

I have problems in understanding the proof that the root spaces of a semisimple Lie algebra are all 1-dimensional and that the only multiples of a root $\alpha \in \Phi$ which occur in $\Phi$ are $\pm ...
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Compactness of semisimple Lie algebra

I want to prove that on a semisimple Lie algebra $\mathfrak{g}$ over ${\bf R}$: $\mathfrak{g}$ is compact if and only if the Killing form is strictly negative definite. Here the Lie algebra is ...