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.
357
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Is it true that for a semisimple Lie algebra $\mathfrak{g}$, $[\mathfrak{g},\mathfrak{g}^e]\subset[e,\mathfrak{g}]$ for $e$ a regular nilpotent?
Let $\mathfrak{g}$ be semisimple Lie algebra, let $e \in \mathfrak{g}$ be a regular nilpotent element ie. an element whose adjoint endomorphism is nilpotent and of maximal rank. Denote by $\mathfrak{g}...
- 11
4
votes
1
answer
91
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Two alternative descriptions of $\mathrm{so}_{2n}(\mathbb{C})$
It seems that there are two ways to define the Lie algebra $\mathrm{so}_{2n}(\mathbb{C})$. The first one is
$\mathrm{so}_{2n}(\mathbb{C})_{(1)}:=\{M \in \mathrm{gl}_{2n}(\mathbb{C}) \ | \ M + M^t = 0\}...
- 149
2
votes
1
answer
63
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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). ...
- 221
2
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56
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Vectors belonging to each component of decomposition of semi-simple Lie group representation
Say that $W$ is a $SL_4(\mathbb{Q})$ representation that decomposes into irreducibles (using the notation of Fulton-Harris) $$W = \Gamma_{a,b,c} \oplus \Gamma_{d,e,f}$$
Now say that I have a vector $v ...
- 55
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1
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58
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$G$-equivariant map between Lie group representations
Say $f: A \rightarrow B$ is a $G$-equivariant map between finite dimensional $G$-representations, for some semi-simple Lie group $G$. If I know:
$B$ decomposes into irreducibles as $B = B_1 \oplus ......
- 55
0
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69
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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 ...
- 55
3
votes
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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 ...
- 481
1
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1
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47
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Do all maximally noncompact Cartan subalgebras in a real semisimple Lie algebra arise from split toral subalgebras?
Let $\mathfrak{g}$ be a real, semisimple Lie algebra and $\mathfrak{g}_{\mathbb{C}} = \mathfrak{g} + i\mathfrak{g}$ its complexification. Following Knapp, a Cartan subalgebra of $\mathfrak{g}_{\mathbb{...
- 1,634
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Help using LiE computer package? [closed]
I'm trying to decompose some tensor products of semi-simple Lie algebra representations into irreducibles. I've read great things about Lie but I get a "502 Bad Gateway" when I use the ...
- 55
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Are the weights of an irreducible representation of a complex semisimple algebra induced by the weights of a representation of $SL(N)$ for some $N$?
If we consider the weights of a complex irreducible representation $V$ of a complex semisimple Lie algebra $\mathfrak{g}$, with respect to a chosen Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$, ...
- 5,186
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141
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Kernel of an irreducible representation on the direct sum of two semisimple Lie algebras
Suppose $L=L_1\oplus L_2$, where $L_1,L_2$ are finite-dimensional complex semisimple Lie algebras. Suppose $V$ is a finite-dimensional irreducible representation of $L$ (through $\rho:L\to \mathfrak{...
- 159
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votes
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65
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Generators and relations for $\mathfrak{sl}(N, \mathbb{C}[t])$
The Lie algebra $\mathfrak{sl}(N, \mathbb{C}[t])$ can be thought of as matrix valued polynomials. As a vector space
$$\mathfrak{sl}(N, \mathbb{C}[t]) = \bigoplus_{k =0}^{\infty}\mathfrak{sl}(N, \...
- 795
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44
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Orthogonal complement of a parabolic subalgebra with respect to an invariant form
Let $F$ be a characteristic $0$ field, with $\mathfrak{g}$ a semisimple Lie algebra defined over $F$. Let $B(\cdot,\cdot)$ be a nondegenerate symmetric bilinear form on $\mathfrak{g}$ which is ...
- 1,353
1
vote
1
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Adjoint of Translation Functors in Category O
Let $\mu,\lambda\in\mathfrak{h}^*$ with $\mu-\lambda\in \Lambda$ and let $pr_\mu$ and $pr_\lambda$ be natural projections of category $O$ onto blocks $O_{\chi_\mu}$ and $O_{\chi_\lambda}$ respectively....
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Root space decomposition of Lie subgroups
Let $\varphi \colon G \to \tilde{G}$ be a group homomorphism between two real semisimple Lie groups. For example, $\varphi$ could be an inclusion of a subgroup $G \subseteq \tilde{G}$. Let $\mathfrak{...
- 1,634
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66
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Lie subgroup of non-abelian compact Lie group is compact?
I need to decide if this statement is true or false:
" Every Lie subgroup of non-abelian compact Lie group is compact."
I think that it is false. I thought in a counterexample in which the ...
1
vote
1
answer
67
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Dynkin diagram of a semisimple Lie algebra
Let $J=\begin{pmatrix}0&0&0&0&0&1 \\ 0&0&0&0&1&0 \\ 0&0&0&1&0&0 \\ 0&0&1&0&0&0 \\ 0&1&0&0&0&0 \\ 1&...
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Integrating Homomorphisms of Borel Subalgebras
Let $G$ be a connected simple complex Lie group and $\mathfrak{g}$ be its Lie algebra. Let us fix a root decomposition, let $\mathfrak{b}_\pm$, $\mathfrak{n}_+$ and $\mathfrak{h}$ be the corresponding ...
- 149
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votes
2
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65
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Non-semisimple representation of vector spaces and of Lie Algebras
Is it correct to say that a non semisimple representation of a vector space is a representation which is reducible and indecomposable?
If yes, how can this intuition be translated to the definition of ...
- 184
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52
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Invariants of graded space is the same as the graded space of invariants
I am stuck on Theorem $3.5.1$ of the following book. The author says
"In particular, we see that the map $(3.5.1)$ is compatible with the natural filtrations, and the corresponding map of ...
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85
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Killing form of a Lie algebra of matrix
Let $L$ be a Lie algebra (with field $F$). Consider the killing form
$$\begin{align*} k:&L \times L \to F \\
&(x,y) \to Tr(ad(x)ad(y))
\end{align*}$$
We know that $L$ is semisimple if and only ...
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0
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prove that a Lie algebra is semisimple [duplicate]
Let
$$J_3=\begin{pmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{pmatrix}$$
$$J=\begin{pmatrix}
0_3 & J_3 \\
J_3 & 0_3
\end{pmatrix}$$
Consider the Lie algebra $L=\{...
- 558
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0
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43
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Lie group of a complex simple Lie algebra $\mathfrak{g}$
The context: I have a complex simple Lie algebra $\mathfrak{g}$. The book I'm reading states
"Let $G$ be the connected simply-connected Lie group with Lie algebra $\mathfrak{g}$."
What is ...
- 41
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1
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If a Lie algbera is not semisimple, why must $\text{ad}_X$ and $\text{ad}_Y$ be of this form?
Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{a}$ a non-zero abelian ideal in $\mathfrak{g}$, and $\mathfrak{s}$ a vector space such that
$$\mathfrak{g} = \mathfrak{a} \oplus \mathfrak{s}.$$
Let $X \...
- 4,721
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45
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A question related to maximally non-compact Cartan subalgebra.
I was looking into the classification of real simple lie algebras from Araki (1962). In page $4$, proposition $1.1$, a criteria is given as to when $\mathfrak{h}^-$ is maximal abelian in $\mathfrak p$ ...
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Proof the image of a Cartan subalgebra under the quotient map is a Cartan subalgebra
Let $g$ be a lie algebra over an algebraically closed field of characteristic $0$. Let $h$ be a Cartan subalgebra of $g$. Let $n$ be an ideal of $g$. Consider the quotient map $q:g \rightarrow g/n$ ...
- 31
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Reference request, relation algebraic groups, real groups and their Lie algebras
Let's work in characteristic $0$. Let $G$ be a semi-simple algebraic group defined over a subfield of $\mathbb{R}$ (invariant under transposition). Let $\mathfrak{g}$ be its Lie algebra (Zariski-...
- 1,634
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2
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Non-reductiveness of upper trianguler matrices $\mathfrak{t}_{n}$ as a Lie algebra
I am having difficulty with a rather simple proposition.
Let $\mathfrak{t}_{n}$ constitute the $n \times n$ upper triangular matrices over $\mathbb{C}$ with the usual matrix commutator, and $\mathfrak{...
- 25
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0
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Irreducible factors of universal enveloping algebra
Let $\mathfrak{g}$ be a simple complex Lie algebra and $U$ be its universal enveloping algebra. We have an action $\mathfrak{g} \to \mathfrak{gl}(U)$ by extending the adjoint action of $\mathfrak{g}$ ...
- 2,878
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47
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Minimal subset of Lie algebra generators required to complete the algebra via the bracket?
Let's say I have a subset of the generators for a real, semisimple Lie algebra. To find the full set of generators, I can take the Lie bracket of every element in my subset until it becomes closed. ...
- 11
0
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$\mathfrak{sl}(3,\mathbb{C})$ irreducible decomposition of a symmetric tensor product space
Let $\mathbb{V}_{1,1}$ the irreducible representation of $\mathfrak{sl}(3,\mathbb{C})$ with higest weights (1,1). I'm asked to find the decomposition in irreducibles of $S^2 (\mathbb{V_{1,1}})$ where $...
- 745
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0
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The Schur multiplier of a Lie algebra
Like with groups, the Schur multiplier of a Lie algebra can be defined in several ways, depending on the context and the specific definitions used. Here are three commonly used definitions:
$(1)$ ...
- 1,878
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0
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72
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The Killing form of a Lie algebra is $DER(\mathfrak{g})$-invariant
The Killing form of a Lie algebra is defined as
$$
B(x,y)=\operatorname{Trace} (\operatorname{ad}_x \circ \operatorname{ad}_y)
$$
I'm trying to show that, given a derivation of the Lie algebra, i.e. $\...
- 745
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0
answers
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$\mathfrak{sl}(2,\mathbb{C})$-Invariant tensor associated to irreducible representations
I'm trying to understand the $\mathfrak{sl}(2,\mathbb{C})$ irreducible representation.
It is known that any irreducible representation $\pi : \mathfrak{sl}(2,\mathbb{C}) \xrightarrow{} GL(V)$, with $V$...
- 745
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0
answers
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A sum over roots of a simple Lie algebra
The following question originates from analyzing the modular $S$-formation of a modular differential equation that annihilates the character of a chiral algebra. A term pops up that is proportional to ...
- 637
2
votes
1
answer
88
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How to calculate $\text{End}(V^{\otimes n})$
Let $\mathfrak g$ be a complex semisimple Lie algebra, and $V$ the fundamental $\mathfrak g$-module. Then we can decompose $V^{\otimes n}$ into the direct sum of irreducibles.
For example, in the case ...
- 753
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Proof that if $L \subset \mathfrak{sl}(V)$ then $L$ is semisimple [duplicate]
I am trying to show that if $V$ is finite dimensional over $\mathbb{C}$, and we have a Lie algebra $L \subset \mathfrak{sl}(V)$ such that the natural representation $(V, \rho:L \subset \mathfrak{gl}(...
- 1,808
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Proving that a symmetric bilinear form on $L$ is invariant
Let $L$ be a Lie algebra and $\rho: L \to \mathfrak{gl}(V)$ a finite-dimensional representation of L. Define a symmetric bilinear form on $L$ by $$\langle x, y \rangle_{\rho} = tr (\rho(x) \rho(y))$$
...
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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 ...
- 23.1k
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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 $S_{\alpha}$...
- 2,142
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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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0
answers
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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 ...
- 745
1
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0
answers
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Sublattice of the weight lattice
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 $\...
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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 + \...
- 2,802
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answer
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Why does the definition of cocyle in a central extension of a Lie algebra work?
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 ...
- 41
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votes
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44
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Show that roots are preserved by $\tau.$
Let $L$ be a finite dimensional semisimple Lie algebra with Cartan subalgebra $H.$ Let $\Gamma$ be the set of all fundamental roots or simple roots of $L.$ Let $\Gamma_1, \Gamma_2 \subset \Gamma$ and $...
- 2,802
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votes
1
answer
91
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specific long exact sequence of derived functors in category $\mathcal O$
I am currently reading Gaitsgorys notes on Category $\mathcal O$ and geometric representation theory (https://people.math.harvard.edu/~gaitsgde/267y/catO.pdf) and I'm having some trouble with the ...
- 172
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votes
1
answer
131
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Root space decomposition of a semisimple Lie algebra.
Let $L$ be a semisimple Lie algebra. I am trying to understand root space decomposition of $L$ on my own. Since $L$ is semisimple, $L$ possesses an abelian maximal toral subalgebra i.e. an abelian ...
- 2,802
4
votes
1
answer
197
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Finite order automorphisms of semi simple lie algebras (Kac Lemma 8.1)
I am currently reading Kac's book on infinite dimensional Lie algebras and have some trouble with Lemma 8.1. The setup is as follows:
Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra ...
- 172