For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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1answer
107 views

Radical of $\mathfrak{gl}_n$

I find it intuitive enough that the radical of $\mathfrak{gl}_n\mathbb F$ is the scalar matrices, but I have trouble finding an easy, but complete proof: Proof. Let $\mathfrak s$ denote the scalar ...
1
vote
1answer
212 views

Is this distribution involutive?

For two days I've been trying to show the following: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and consider the smooth distribution $$F=\{F_p=DR_p(e)\mathfrak{h}; p\in G\},$$ where ...
2
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1answer
346 views

Solvable Lie algebra with codimension 1 ideal

There is an exercise in Humphreys's An Introduction to Lie Algebras and Representation Theory: "Any nilpotent Lie algebra contains a codimension 1 ideal". The proof I am thinking of is the following. ...
2
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1answer
213 views

Weyl group and Dynkin diagram

Can somebody help me with following questions: 1)Prove that two simple roots in a Dynkin diagram that are connected by a single edge are in the same orbit under the Weyl group. and 2)For an ...
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1answer
100 views

Question about lie bracket..

Let $G$ be a Lie group with Lie algebras $\mathfrak{g}$ and let $\mathfrak{h}\subseteq \mathfrak{g}$ be a Lie subalgebra. Write $F_p=DR_p(e)\mathfrak{h}$, $p\in G$, where $R_p:G\rightarrow G$ given by ...
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0answers
99 views

adjoint representation completely reducible

Let $\mathcal{A}$ be a Lie algebra. Suppose that adjoint representation of $\mathcal{A}$ is completely reducible (or semisimple). Show that $\mathcal{A}$ can be written as a direct sum of semisimple ...
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1answer
90 views

Confusion regarding the augmentation map of Lie algebras

Let $\mathfrak{g}$ denote a Lie algebra, $K$ a field, and view $K$ as a trivial $\mathfrak{g}$-module (that is, define $x \cdot a = 0$ for all $x \in \mathfrak{g}, a \in K$). In other words, we have a ...
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1answer
80 views

Equality involving Lie Brackets

I have a question concerning Lie brackets: Consider the Lie bracket $$[, ]:\mathfrak{g}\times \mathfrak{g}\rightarrow \mathfrak{g},$$ where $\mathfrak{g}=T_eG$ is the Lie algebra of a Lie group $G$. ...
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1answer
165 views

The Lie algebra of the commutator subgroup

If $G$ is a connected Lie group with Lie algebra $g$, then is its commutator subgroup $[G,G]$ a closed subgroup with Lie algebra $[g,g]$?
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1answer
257 views

Lie bracket of vector fields definition equivalence

Lie bracket of vector fields is defined in two ways: Let $\Phi^X_t$ be the flow associated with the vector field $X$, and let $d$ denote the tangent map derivative operator. Then the Lie ...
4
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1answer
117 views

Same Dynkin diagrams $\Longrightarrow$ Isomorphic root systems.

I am studying the book Introduction to Lie algebras. In page 122 there is something I don't understand and I am looking for some help. In the beginning of that page the authors give the definition of ...
0
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1answer
104 views

What is the number of non-compact generators of $\operatorname{so}(p, q)$ and $\operatorname{su}(p, q)$?

Setting $n = p + q$, the total number of generators of $\operatorname{so}(p, q)$ or $\operatorname{su}(p, q)$ is respectively $n(n - 1) /2$ and $n^2 - 1$. But what is the number of non-compact ...
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1answer
177 views

Automorphism group of Lie algebra $\mathfrak{g\oplus g}$

Let $\mathfrak g$ be the Lie algebra of strictly upper triangular 3x3 matrices. How can I determine the group $\operatorname{Aut}(\mathfrak{g\oplus g},\Delta\mathfrak g)$, where $\Delta\colon\mathfrak ...
2
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0answers
165 views

Determinant of a cartan matrix

I was taking an introductory course in Lie algebras and I just learned about how we associate a Cartan matrix to a semisimple Lie algebra. So, for the A-series, the determinant of this matrix goes to ...
2
votes
1answer
582 views

Tangent space at the identity element of a lie group

Let G be a lie group . we know a Lie group is a group with a smooth manifold structure s.t both the multiplication map $m$ and group inversion map $i$ are smooth . Now by identifying ...
3
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1answer
242 views

Definition of lie bracket of vector fields

The Jacobi-Lie bracket or simply Lie bracket, $[X,Y]$, of two vector fields $X$ and $Y$ is the vector field such that $[X,Y](f) = X(Y(f))-Y(X(f)) \,.$ ...
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1answer
111 views

Isomorphisms of the Lorentz group and algebra

I'm trying to read a few books on QFT and some seem to say the Lorentz algebra obeys $\mathfrak{so}(1,3)\otimes \mathbb{C} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ while others say ...
3
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2answers
483 views

Decomposing tensor product of lie algebra representations

I'm given a lie algebra representation $\pi$ of some semi-simple algebra and that it decomposes into a sum of irreducible representations. What technique should I use to show the decomposition of ...
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1answer
38 views

concerning coadjoint representation

Let $\xi $ be the vector field on $\frak{g}^*$ (dual of Lie algebra) which correspond to element $X$ of the Lie algebra $\frak{g}$. Then why have we $\xi(F)=K_*(X)F$ where here $K=Ad^*(g)$ is ...
11
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2answers
463 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
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1answer
285 views

Classifying all rank 2 and 3 root systems

I am working with the representation theory of complex simple Lie algebras, and have a question: It is intuitively clear that the root systems $A_1\times A_1$, $A_2$, $B_2$, and $G_2$ comprise all ...
0
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1answer
92 views

Lie bracket of vector fields on $\Bbb R^{n}$

Please show how to solve? I am stack with lie bracket. Thank you.
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1answer
117 views

Why is the dual space of Cartan subalgebra an irreducible representation of Weyl group

it is proposition 14.31 in Fulton-Harris book. The proof goes like this. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and assume $\mathfrak{z}\subseteq\mathfrak{h}^*$ were preserved ...
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0answers
63 views

Usage and determination of “rank” and “dimension” of groups & representations

Physicist here. I seem to see conflicting statements about the rank of some groups I've come across lately. A paper I'm reading states that $SO(6)$ is rank 3 and therefore its Cartan subalgebra ...
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1answer
102 views

Is algebraic closure required in Weyl's theorem on complete reducibility? (Lie algebras)

Weyl's theorem states that finite-dimensional representations of finite dimensional semisimple Lie algebras are completely reducible (expressible as a direct sum of irreducible submodules), with some ...
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1answer
55 views

Is this true? $\forall z \in sl(2) \;\exists\,x,y \in sl(2)\,;\,z=[x,y]$

A semi simple Lie Algebra (LA) $\mathbf{g}$ is usually defined as (I) a direct sum of simple LAs: $\mathbf{g}\,=\,\oplus_i\,\mathbf{g_i}$. An alternative characterization seems to be the statement ...
2
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1answer
106 views

Lie algebra and lie brackets - inconsistent?

In http://en.wikipedia.org/wiki/Lie_algebra, Lie bracket operation is defined as having bilinearity: $[ax+by,z] = a[x,z]+b[y,z], [z,ax+by] = a[z,x]+b[z,y]$. But in ...
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1answer
53 views

Why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?

Let $G=SO(n,R)$ be a Lie group and $\mathbb{g}$ its lie algebra. Take $X\in \mathbb{g}$. Then why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?
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1answer
66 views

Lie group of orthogonal matrix group - what would this mean?

In Lie group terms, this means that the Lie algebra of an orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric ...
0
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1answer
213 views

Real representations of Lie algebra $\mathfrak{so}(3)$

How does one construct an $n$-dimensional, irreducible, real-valued and non-zero representation of the three generators of the Lie algebra $\mathfrak{so}(3)$ for a given value of $n$?
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0answers
109 views

Regarding the definition of vector field flow

To make the connection to the Lie derivative, let $t \mapsto \Phi^X_t$ be the 1-parameter group of diffeomorphisms (or flow) generated by the vector field $ X $. The differential $ d\Phi^X_t ...
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0answers
229 views

Centralizer of an element in a Cartan subalgebra is reductive.

Let $\mathfrak{g}$ be a Lie algebra with Cartan subalgebra $\mathfrak{h}$ and root system $\Phi$. Show that $C_\mathfrak{g}(h)$ is reductive, that is $Z(C_\mathfrak{g}(h))=Rad(C_\mathfrak{g}(h))$, ...
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1answer
132 views

The Weyl group preserves inner product

Show that the Weyl group $W$ preserves the inner product: $(w(\lambda)\,,\, w(\mu)) = (\lambda, \mu)$ for all $w\in W$ and $\lambda, \mu\in E$. I know it suffices to check this on reflections ...
8
votes
1answer
152 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
4
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1answer
125 views

Exercise in Erdmann's Intro to Lie algebras

I'm working on question 4.8 on page 36 of Erdmann and Wildon's book called Introduction to Lie Algebras. The question is as follows: Let $L$ be a Lie algebra over a field $F$, such that ...
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1answer
34 views

$x \in \left[L,L\right] \Rightarrow tr(ad \, x)=0$

Suppose $L$ is a Lie Algebra and $x \in L'=\left[L,L\right]$. As a homework problem, I need to show that $\operatorname{tr}(\operatorname{ad} \, x)=0$. I assumed $\dim(L)<\infty$ (not sure if this ...
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vote
2answers
302 views

A covering map from a differentiable manifold

Let $p: C \to X$ is a covering map. Suppose that $C$ is a differentiable manifold. Is X - differentiable manifold? More precisely, I am interested in the case where $C$ is Submanifold of Lie algebra, ...
3
votes
1answer
130 views

Finding the Lie Algebra of a Lie Group

I am having a hard time finding the set of all $X \in M(n, \mathbb{R})$ such that $e^{tX^T}Be^{tX} = B$ for all $t \in \mathbb{R}$. where $b$ is any matrix in $M(n, \mathbb{R})$.
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1answer
297 views

Lie Groups induce Lie Algebra homomorphisms

I am having a difficult time showing that if $\phi: G \rightarrow H$ is a Lie group homomorphism, then $d\phi: \mathfrak{g} \rightarrow \mathfrak{h}$ satisfies the property that for any $X, Y \in ...
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1answer
272 views

A quesion in Fulton & Harris book “representation theory a first course”

In Section 11.2 A little plethysm, it discusses the tensor product of two different representations of $sl_2\mathbb{C}$. It says "If $V=\bigoplus V_{\alpha}$ and $W=\bigoplus W_{\beta}$ then ...
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1answer
80 views

Embedding of $PGL_n\mathbb{C}$ and friends

I would like the find an embedding/faithful representation from the projective linear group $PGL_n\mathbb{C}\to GL_m\mathbb{C}$ for some $m$, and likewise for the other projective groups ...
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1answer
96 views

Question about Lie Groups

I am having trouble with the following Lie Algebra question. I will appreciate any help greatly. Any Lie group homomorphism $\phi : G \rightarrow H$ is determined by the induced Lie algebra ...
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0answers
24 views

Finding the Lie map

Suppose I have a group homomorphism $\rho:SL(2,\mathbb{C})\to SO_0(3,1)$ given by $\rho(a)X=aXa^*$ and I want to see how the corresponding Lie map $L\rho$ looks like. By definition $$ ...
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1answer
126 views

Isomorphism of Lie algebras as similarity transformation

If two finite dimensional matrix Lie algebras are isomorphic, is it always possible to see the isomorphism as a similarity transformation $g \mapsto M^{-1} g M$ ?
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2answers
88 views

Simple Lie Algebras and its derived series

Trivially, for any Lie Algebra (LA) g, g':=[g,g] is an ideal. What's wrong with the following argument? Be g a simple LA, then it has to be g'=g by definition of simple LA. But [g,g]=g seems to be an ...
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0answers
146 views

Exterior and symmetric powers of $\mathfrak{sl}(4,\mathbb{C})$ representation

I am taking a course on representation theory, and going through Lecture 15 of Fulton and Harris's Representation Theory. One of the topics we're currently covering is the example of ...
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68 views

Projection map $\text{Sym}^2(\text{Sym}^3V)\to \text{Sym}^2V$ viewed as a Hessian

Exercises 11.21 and 11.22 in Fulton's Representation Theory are the following: Let $V$ be the standard representation of $\mathfrak{sl}_2\mathbb{C}$. The projection map from ...
2
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1answer
92 views

Connectedness of a normal subgroup of a linear Lie group.

Let $G\leq GL(n,\mathbb{R})$ be a connected linear Lie group and let $N$ be a normal subgroup of $G$ such that their Lie algebras are isomorphic: $$\mathfrak{g}\cong\mathfrak{n}$$ Does it follow that ...
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0answers
32 views

Only 3 finite-dimensional Lie algebra on $\mathbf R$?

Please, how does one show that up to diffeomorphism there are exactly three finite dimensional Lie algebras of vector fields on the real line $\mathbf R$.
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2answers
69 views

Tangent Space of $\operatorname{Aut}(T_e G)$

Let $G$ be a Lie Group, how to prove that the tangent Space of $\operatorname{Aut}(T_e G)$ at identity is $\operatorname{End}(T_e G)$? Thanks.