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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1answer
61 views

When is the Lie bracket of two root spaces nonzero?

Let $L$ be a semisimple Lie algebra with root space decomposition $L = L_0 \oplus \bigoplus_{\alpha \in \Phi} L_\alpha$. For roots $\alpha, \beta \in \Phi$, we always have $[L_\alpha, L_\beta] \...
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52 views

dimension of Ext groups between a Verma module and the tensor product of a module with a finite-dimensional irreducible

Setting: Let us work in the vanilla category $\mathcal O$, let $M_\lambda$ be the Verma of highest weight $\lambda$, with irreducible quotient $L_\lambda$. Let $\mu\in\Lambda_+$ be a integral dominant ...
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15 views

Dimension formula for a semisimple lie group

The attached picture is from Hitchin's Duke paper STABLE BUNDLES AND INTEGRABLE SYSTEMS. Let G be a semsimple complex Lie group and $P_1,P_2...P_k$ be a basis of homogeneous invariant polynomials in ...
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20 views

Is the adjoint map $G^0\to GL(\mathfrak g)$ proper for semi-simple $\mathfrak g$?

Suppose that $G^0$ is the connected component of a semi-simple Lie group. Is the adjoint representation $G^0\to GL(\mathfrak g)$ a proper map? I'm not sure if this is an obvious statement or not. I ...
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55 views

Why find Casimir elements?

Given a semi-simple Lie algebra $\mathfrak g$ the Killing form induces a canonical central element of $U(\mathfrak g)$. This is seen as a very important and useful thing. Why is it useful to know ...
2
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1answer
136 views

Semisimplicity of so(B)

Given a symmetric bilinear form on $\mathbb{C}^n$ (with associated symmetric matrix $B$) and the algebra $so(B)$ of complex traceless matrices $X$ with $XB+BX^T=0$, i am trying to prove the following ...
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3answers
110 views

Fun example of representations of $sl_2(\mathbb C)$

What are fun occurence of $sl_2(\mathbb C)$ representations "in nature"? Ideally elementary examples would be the best. Examples I know : The cohomology ring $H^*(X,\mathbb C)$ for $X \...
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40 views

Weyl group extension as inner automorphisms

Notations: $L$ a semisimple Lie algebra over $\mathbb{C}$ with $\{e_i,f_i,h_i\}_{i=1}^{n}$ the algebraic generators $H$ the Cartan subalgebra $W=\langle s_i|i=1,\cdots,n\rangle$ the Weyl group and $A=...
3
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1answer
67 views

If $A$ is a direct sum of matrix algebra over $C$, what are all finite dimensional simple $A$-modules?

If $A$ is a direct sum of matrix algebra, what are all finite-dim simple $A$-modules? For simple case of $2$ matrix algebras, consider $A = M_n(\mathbb{C}) \oplus M_m(\mathbb{C})$. Suppose $M$ is a ...
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31 views

Isomorphisms of maximal toral subalgebras

Take $L$ a semisimple complex Lie algebra and $H$ a maximal toral subalgebra. I want to prove that given a second maximal toral subalgebra $H'$ then exists an inner automorphism of $L$, $\omega$, such ...
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17 views

Is $\mathfrak{so}_{\mathbb{R}}(p,q)$ semisimple? [duplicate]

Consider the real Lie algebra $\mathfrak{so}(p,q)$ with $p\geq q\geq 1$. Question: For what pairs $(p,q)$ is this Lie algebra semisimple? Part of answer: Obviously $\mathfrak{so}(2,0)$ and $\mathfrak{...
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Calculation of Dynkin Index issue

Following Onishchik's "Topology of Transitive Transformation Groups" section 3.10 the Dynkin index, $j_{\rho}$, of a Lie group homomorphism $\rho: H \rightarrow G$ of connected simple ...
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1answer
130 views

Prove Harish-Chandra's theorem on central characters with algebraic-geometrical observation and the linkage relations among weights

For representations of semisimple Lie algebras $L$, we have the following theorem on central characters: Theorem (Harish-Chandra): Let $\lambda, \mu \in H^{\ast}$. Then $\chi_{\lambda} = \chi_{\mu}$ ...
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Exercise 24.12 in Humphrey's Lie algebra book : Application of Steinberg's formula

I'm working on the following exercise (Exercise 24.12) in Humphrey's book Introduction to Lie algebras and representation theory. My question is how to solve the following exercise: Exercise: Deduce ...
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40 views

Irreducible finite-dimensional real $\mathfrak{sl}_2(\mathbb R)$-representations

It is easy to show that every irreducible finite-dimensional real $\mathfrak{sl}_2(\mathbb C)$-representation is a weight module. The operator commonly denoted as $h$ in $\mathfrak{sl}_2$ has an ...
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75 views

A counterexample to Weyl theorem on complete reducibility

A well known result from Lie algebras' representations is the Weyl Theorem (Theorem 6.3 of Humphreys' Introduction to Lie Algebras and Representation Theory), which states the following: Let $\phi: \...
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0answers
41 views

Computation of finite dimensional irreducible representations of non compact reductive Lie groups with finite number of connected components

Let $G$ be a reductive (complex or real) Lie group with a finite number of connected components. Can one construct an algorithm to generate any finite-dimensional representation $\rho$ of $G$ (meaning ...
3
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2answers
188 views

Every automorphism preserves a Cartan subalgebra is induced by an element in the Weyl group

Let $\mathfrak{g}$ be a seimisimple Lie algebra over $\mathbb{C}$, $\mathfrak{h}$ a Cartan subalgebra, $\mathrm{Aut}^0(\mathfrak{g})$ the connected component of the identity in the automorphisms group ...
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1answer
44 views

Isomorphisms of irreducible root systems

Let $E,E'$ be two euclidean vector spaces and $\Phi,\Phi'$ two root systems of $E$ and $E'$, respectively. Let $\varphi:E\to E'$ be an isomorphism of root systems. Applying the definition of root ...
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1answer
35 views

The kernel of $\operatorname{ad}_x = [x, \cdot]$ consists of 0 and ${\rm span}\{x\}$, when is this true?

The Lie algebra $\mathfrak{s}\mathfrak{o}(3)$, when identified as vectors in $\mathbb{R}^3$ with the cross product as the bracket, has the property that the only $y \in \mathfrak{s}\mathfrak{o}(3)$ ...
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45 views

What does it mean for a root to be an element of the Lie algebra?

My understanding of roots in Lie algebra theory is as follows: given a Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ with basis $\{H_i\}_{i=1}^{r}$, and root vectors $\{X_\alpha\}_\alpha$, the ...
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0answers
31 views

Discrete series representation of SO(2,1)

Since so(2,1) is a noncompact Lie algebra, its unitary representations are all infinite-dimensional. Now, I have seen that the unitary representations of so(2,1) can be divided into three categories: ...
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28 views

Prove that L is a semisimple algebra [duplicate]

Consider the algebra $L=\mathfrak{so}(6,\mathbb{C})=\{x\in\mathfrak{gl}(6,\mathbb{C}): x^tJ+Jx=0\}$, where $J_3=\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{...
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1answer
42 views

Is the Lie algebra so(2,6) semi-simple?

I'd like to know whether the Lie algebra so(2,6) is semi-simple. I know that I could compute the Killing form for that. But is there an easier way?
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1answer
39 views

A question about roots and root spaces of semisimple Lie algebras

In the book "Lie Groups: Beyond and Introduction" by Anthony Knapp, the author describes roots of a semisimple (A Lie algebra without any non-zero solvable ideal) Lie algebra as elements of ...
4
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1answer
80 views

Softwares to determine semi-simple types of Lie algebras generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of matrices

I wish to determine the type of Lie algebra generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of square matrices with irrational elements. I've been using GAP ...
2
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1answer
85 views

A description of the $A_n$-root system in terms of $\mathbb{R}^{n}$

The root system $(V,\Delta)$ of type $A$ is usually defined as a subset of $\mathbb{R}^{n+1}$: $$ \{ e_i - e_j ;\, 1 \leq i,j \leq n+1,\, i \neq j\}. $$ However these vectors span an $n$-dimensional ...
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34 views

Character formula for real representations

For an irreducible representation of a complex semisimple Lie algebra the Weyl character formula is well known. For an irreducible real representation of a real semisimple Lie algebra is there a Weyl ...
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0answers
35 views

Scaling behaviour and volumes in coadjoint orbits

Let $\mathfrak{g}$ be a semisimple real Lie algebra and $G$ its associated adjoint Lie group. If we set $\kappa$ to be its Killing form, and $\theta$ to be some Cartan involution, we get a scalar ...
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0answers
20 views

Constructing a surjective intertwining map onto Verma module

I want to prove the following result: Theorem: Let $\mathfrak{g}$ be a semi-simple Lie algebra, $\mathfrak{h}$ be a Cartan subalgebra, $\mu \in \mathfrak{h}$, and $W_{\mu} = \mathfrak{U}_{\mathfrak{g}...
2
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1answer
92 views

Compact real form of Semisimple Lie algebra

I am reading B.Hall's book "Lie groups, Lie algebras, and Representations". You can find there a definition for semisimple Lie algebra. DEF1: A complex Lie algebra $\mathfrak{g}$ is ...
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1answer
20 views

Exactness of tensor product with finite dimensional g-module in category $\mathcal O$

One of the properties of the category $\mathcal{O}$(w.r.t a finite dimensional semisimple Lie algebra $\mathfrak{g}$) is: If $0\to M_1\to M_2\to M_3\to 0$ is an exact sequence of objects of $\mathcal{...
3
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1answer
53 views

The action of the longest element on weights

I am doing some computations related to some work I do in Lie theory and I need to compute the result of the action $$w_0 \omega_{i},$$ where $w_0$ denotes the longest element in the Weyl group $W$ of ...
2
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1answer
164 views

Does every nilpotent element in a real Lie algebra lie in some minimal parabolic Lie algebra?

Definitions: Let $\mathfrak{g}$ be a finite-dimensional, real, semisimple Lie algebra with some Cartan involution $\theta$. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the usual Cartan ...
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1answer
202 views

On a simple real Lie algebra whose complexification is not simple

I wish to prove the following result: Theorem: Let $\mathfrak{h}$ be a real simple Lie algebra and $\mathfrak{h}_{\mathbb{C}}$ be its complexification, which is not simple. Then, there is a complex ...
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0answers
37 views

What Does It mean to Solve the Lie Algebra

I have found the Lie Symmetries for a particular PDE, and have constructed the commutator table. The symmetries are as follows: \begin{align} X_1 &= x \frac{\partial}{\partial x} + 3t \frac{\...
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0answers
33 views

Kernel of the quotient adjoint representation for a parabolic subgroup

Let $P$ be a complex parabolic subgroup of a semi-simple complex group $G$. Then we have a decomposition $\mathfrak{g}=\mathfrak{g}_{-k}\oplus \ldots \oplus \mathfrak{g}_0 \oplus \ldots \oplus \...
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1answer
42 views

Casimir element of a semisimple Lie algebra is "additive"

Definitions Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be semisimple Lie algebras, and suppose that $(V,\rho)$ is a representation of $\mathfrak{g} = \mathfrak{g}_1 \oplus \mathfrak{g}_2$. Given a ...
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0answers
70 views

Why is the quotient by the Borel subgroup compact?

Let $G$ be a complex lie group with semisimple lie algebra $g$. Put $b$ the borel subalgebra (so the sum of $h$ and all the positive roots) for some choices. One can there is the borel subgroup $B$ ...
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0answers
67 views

Representation of reductive Lie algebras

Let $\mathfrak{L}$ be a reductive Lie algebra. Suppose that $\mathfrak{L}$ is a direct sum of $\mathfrak{g}$ and $\mathfrak{h}$ as Lie algebras where $\mathfrak{g}$ is a semisimple Lie algebra and $\...
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1answer
71 views

Decompose weights of same multiplicity into different Weyl group orbits

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$. Let $\lambda\in\mathfrak{h}^*$ be a dominant integral ...
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0answers
36 views

Which real Lie group is the stabilizer of this Hermitian form?

Let $h\colon \mathbb{C}^{2n} \times \mathbb{C}^{2n} \rightarrow \mathbb{C}$ be the form $$h(\vec{v},\vec{w}) = \left(v_1 \overline{w}_1 + \cdots + v_n \overline{w}_n\right) - \left(v_{n+1} \overline{w}...
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2answers
71 views

Relation between the dimension of a Lie Algebra and the dimension of its complexified version

Suppose $L$ is a semi-simple Lie Algebra with dimension $d$. The complexification of the Lie Algebra over $\mathbb{R}$ is given by $$L_{\mathbb{C}} = L \otimes_{\mathbb{R}} \mathbb{C}$$ My question is:...
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0answers
107 views

Computing the rank of the Lie Algebra of the symplectic group

I was unsure if I should post this to the physics forum since mathematician often tend to think differently than physicists, but this is a claim about Lie algebras so I thought this forum might be ...
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0answers
37 views

How to generate an associative algebra given a Lie subalgebra

I am new to the study of Lie algebras and particularly algebraic Lie algebras. In the paper Chevalley, Claude, Algebraic Lie algebras, Ann. Math. (2) 48, 91-100 (1947). ZBL0032.25202. the author ...
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95 views

When are multiplicities of two weights equal in simple highest weight modules

Let $\mathfrak{g}$ be a semisimple (or simple) Lie algebra over $\mathbb{C}$. Take a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. Let $\Phi$ be the root system of $\mathfrak{g}$ w.r.t. $\...
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1answer
47 views

Cartan-Weyl basis in terms of simple roots

Let $\mathfrak{g}=\mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi}\, \mathfrak{g}_\alpha$ be a complex semisimple Lie algebra, where $\mathfrak{g}_{\alpha}=\mathbb{C} \cdot e_\alpha$ is 1-dimensional. ...
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0answers
50 views

Nilpotent orbits from either Dynkin diagram or Levi subalgebras

Given a nilpotent orbit, $\mathcal{O}_X$, associated to an element, $X$, of a complex semisimple Lie algebra, $\mathfrak{g}$, there are two equivalent classification schemes: Via the Jacobson-Morozov ...
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1answer
45 views

Weyl group $W$ is a normal subgroup of $Aut(R)$

Let $R$ be a root system and $W$ be the Weyl group of $R$. Then we have that the group $W$ is a normal subgroup of the group $Aut(R)$ of automorphisms of $V$ ($V$ a finite dimensional vector space) ...
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0answers
31 views

Two roots are proportional in this case

In the book of Serre's "Complex Semisimple Lie Algebra", we found the following on the page 29: Can we rescale the image on math.stackexchange? I don't understand there the last part. How ...

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