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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$\mathfrak{sp}_4$ is a subspace of the vector space of all $4\times 4$ matrices

Let $\mathfrak{sp}_4$ denote the set of all matrices $X$ satisfying $$X^TM+MX=0$$ How can I show that $\mathfrak{sp}_4$ is a vector subspace of the vector space of all $4\times 4$ matrices? I ...
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32 views

İf x is diagonalizable then ad(x) is also diagonalizable

I start to study lie algebras from K. Erdmann, Mark J. Wildon-Introduction to Lie Algebras and i try to solve question below but actually i can't see .How can i start ? Give me hint please Let ...
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exercise in humphrys lie algebra book

show that there exist a unique w∈ W such that wΔ=-Δ.show further that reduced expansion of w involves all simple roots.what is l(w)?
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exercise 19 section 10 in humphrys lie algebra

W is the Weyl group. A subset Δ⊆Φis called base if Δ={α1,α2,.....αn} 1.Δ is a basis of R^n. 2.every α∈Φ can be written as α=k1α1+.......+knαn,ki∈Z+ ∀i or -ki∈Z+ ∀i. elements of Δ are called simple ...
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25 views

Nested commutators that don't vanish

So I've been reading up on Lie Groups and Lie Algebras and the Baker-Campbell-Hausdorff formula. I understand how the formula works and that most of the time the nested commutators vanish at a certain ...
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29 views

Why for simple roots in Lie algebras the master formula reduces to one integer?

The master formula for two generic weights (roots) is $$ 2 \frac{\vec{a} \cdot \vec{b} }{\vec{a} \cdot \vec{a} }=q-p $$ but if we require that the roots are simple then this reduces to $$ 2 ...
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1answer
11 views

Need definition of symmetric and antisymmetric tensor representations of a Lie algebra

I couldn't find a definitive answer online. Suppose we have a representation of a Lie algebra $(\pi,V)$. Consider the symmetric and antisymmetric vector subspaces of the $k$-th tensor product of ...
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34 views

Clarification on notation of “left invariant fields” (Lie groups)

In these notes in Definition 1.4 we learn that A vector field $X$ on a Lie group $G$ is called left invariant if $d(L_g)_h(X(h))=X(g(h))$ for all $g,h \in G$, or for short $(L_g)_*(X)=X$. where ...
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59 views

Normal Subgroups of $SU(n)$

I was wondering if there is any classification for normal subgroups of $SU(n)$? In particular, I think that the answer is no for $n = 2$ by looking at the covering map onto $SO(3)$, but I was curious ...
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Help! How to derive the result related to Darboux derivative?

First, define Darboux derivative. There is one Lie group $G$ and one manifold $M$. Let $\phi:M\rightarrow G$ be a smooth map. The Darboux derivative $\Delta(\phi):TM \rightarrow M\times \mathfrak g$ ...
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Generalization of Schur's Lemma: finite dim. representations of real Lie algebras

Let $V$ be an irreducible finite dimensional real representation of a real finite dimensional Lie algebra $\mathfrak{g}$. From Schur's Lemma, what is $Hom_\mathfrak{g}(V,V)$ or ...
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43 views

Are unipotent algebraic groups connected?

Is a unipotent algebraic group over a field of characteristic zero always connected?. As far as I know, every unipotent algebraic group over field of characteristic zero is isomorphic to a closed ...
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11 views

How to prove $e^{π(ω)} = cos(ω)I_3 + (1 − cos(ω))ωˆ ⊗ ωˆ + sin(ω)π(ωˆ )$?

How to prove $e^{π(ω)} = cos(ω)I_3 + (1 − cos(ω))ωˆ ⊗ ωˆ + sin(ω)π(ωˆ )$, where $ω = |ω|, ωˆ =ω/|ω|.$ My Attempt: In my understanding, $\pi (w)$ is an element of the lie algebra, which is a ...
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22 views

Why is left-invariant vector fields needed to construct a Lie algebra from a Lie group?

Since the set of all vector fields $V$ on a Lie group $G$ forms a vector space, one can impose algebraic structure (a Lie algebra) by defining the bracket $[\cdot,\cdot]$ between these vector fields. ...
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1answer
30 views

Exterior derivatives involving representations

I have two questions regarding the exterior derivative of vector valued forms when representations are involved: Suppose $V$ is a vector space, $M$ a smooth manifold and $\omega$ is a $V$ valued ...
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22 views

The negative of a vector field and its flow

I have a relatively short question about vector fields. Let $G$ be a Lie Group, and $X$ a smooth vector field on it. If its flow is $\left\{\phi_t\right\}$ what is the corresponding flow for $-X$? ...
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Principal bundles, connection forms and fundamental vector fields

Suppose $\pi:P\rightarrow M$ is a principal bundle, $\omega\in \Omega^1(P;\mathfrak{g})$ is the connection one form and $\sigma(\cdot)$ is the fundamental vector field associated to some vector field ...
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19 views

Cohomology group of an algebra

I tried to guess what is a cohomology group of an algebra. I would like to find the correct definition of this. I know what is a cohomology group of a group, but I don't know how connect the second ...
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1answer
64 views

Is there a more intelligent way to compute the determinant of the Killing form of $\mathfrak{sl}(3,F)$?

Is there a more intelligent way to tackle exercise 7 of paragraph 5 of Humphreys (Introduction to Lie Algebras and Representation Theory)? Exercise 7: Relative to the standard basis of $L = ...
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Special linear lie algebra relative with Angular Momentum

Let have the special linear algebra $\operatorname{Sl}(2,\mathbb F)$ ,which is the set of $ 2 \times 2$ matrix with trace zero. I have to prove that the lie algebra $ g=\operatorname{Span}\{ ...
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51 views

How to visualize $\operatorname{Lie}(\operatorname{GL}_n)=\mathfrak{gl}_n$ in positive characteristic?

I'd like an intuitive explanation as to why the Lie algebra of $\operatorname{GL}_n$ is $\mathfrak{gl}_n$ when working over fields of positive characteristic. Below I reproduce how I "see" this fact ...
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When is a stabilizer group a reflection group?

Let $G$ be a compact, connected Lie group and $K$ a closed, connected subgroup. If $K = T$ is a maximal torus, it is well known that $W := N_G(T)/Z_G(T) = N_G(T)/T$ is a finite reflection group, the ...
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1answer
22 views

How to show that set of diagonal matrices is the maximal toral subalgebra of sl(n)

sl(n) is the set of nxn matrices with trace=0. i know that sl(n) is a finite dimensional simple lie algebra and the maximal toral subalgebra of a finite dimensional semi simple lie algebra is abelian. ...
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32 views

How to find subalgebras of standard lie algebras

As I understand it, the symplectic Lie group $Sp(2n,\mathbb{R})$ of $2n \times 2n$ symplectic matrices is generated by the matrices in ...
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1answer
29 views

How to prove that the bond lattice is a lattice w.r.t. the ordering of refinement

Let $G$ be a graph. Let $L_G$ be the bond lattice of $G$ which consists of all partitions of the vertex set of $G$ in which each part induces a connected subgraph of $G$. How to prove that $L_G$ is a ...
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25 views

Where does the $-k\delta$ part of the expression for weights come from?

Consider the affine Lie algebra with the Cartan matrix $$ \left(\begin{array}{cc} 2 & -2\\ -2 & 2 \end{array}\right) $$ Let $\omega_{0}$ be the zeroth fundamental weight, $\alpha_1$ the first ...
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To what extent are the Jordan-Chevalley and Levi Decompositions compatible.

I know that the Jordan-Chevalley decomposition for real Lie algebras only applies to semisimple Lie algebras, but in general the addititive J-C decomposition says that for ANY operator, we can ...
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24 views

Center of the dual of a Lie algebra

Let $\mathfrak{g}$ be a Lie algebra. Let $C \subset \mathfrak{g}^*$ be the subspace of linear forms that vanish on Lie brackets: $$C = \{\alpha \in \mathfrak{g}^*, ~ [\mathfrak{g}, \mathfrak{g}] ...
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1answer
57 views

How to calculate the derivative of a Lie bracket in a coordinate-free setting?

For a given Riemmanian connection defined on a smooth manifold $M$, we denote its covariant derivative by $D_V$ where $V\in \mathcal{x}(M)$, the smooth vector fields on this manifold. Then is it ...
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1answer
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Symplectic lie algebra

Can anyone explain me why, in the symplectic lie algebra, which is defined as $ sp(n)=\{X \in gl_{2n}:X^tJ+JX=0\}$ where $J=\begin{pmatrix} 0 & I \\ -I & 0 \\ ...
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2answers
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Confusion about Lie groups in Fulton & Harris

Near the beginning of chapter 8 (titled Lie groups and Lie algebras) authors motivate the definition of Lie algebra. I'm confused by two things in just one sentence: ($G$ is a Lie group) The ...
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1answer
49 views

Construct representation of Lie algebra

Consider the 3D Lie algebra with the following defined: $ [x,y] = z $, $ [x,z]=0 $, $ [y,z]=0 $ From this, I want to get a little help how to start finding a 3D representation by $3\times 3$ real ...
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1answer
51 views

The curvature of the connection one form - misunderstanding

Let $(P,M,G,\pi,\cdot)$ be a principal bundle. Let $\omega$ be the connection one form for a connection $H\subset TP$. Let $X,Y$ be smooth vector fields on $P$. Then the curvature $\Omega$ of the ...
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1answer
25 views

Advantages of solvability, nilpotency and semisimplicity of Lie algebras?

After pondering on the notion of solvability, nilpotency and semi-simplicity of linear Lie algebras for days (I have been reading Humphreys' Introduction to Lie algebra lately), I remember a professor ...
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Dominant Weight

I am reading a paper which begin by a reminder about root system associated to a simple lie algebra $\mathfrak g$. let $\mathfrak h\subset \mathfrak g$ a cartan subalgebra. Question 1: It says that ...
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1answer
28 views

Definition: What is a two-sided Lie ideal of a Lie algebra?

Let $\mathfrak{g}$ be a Lie algebra and let $\mathfrak{h}$ be a subalgebra. According to wikipedia, $\mathfrak{h}$ is called an ideal of $\mathfrak{g}$ if it satisfies the condition that ...
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80 views

Lie Groups/Exponential map identity

I have come across this identity a few times and I have absolutely no idea why it holds. $g^{-1}\exp(tX)g=\exp(t(\text{ad}_{g^{-1}}X))$ Would any one be able to explain exactly why this holds or ...
2
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1answer
13 views

Finite subgroups which are normal in a matrix Lie group

I have the following question: Let $G$ be a closed subgroup of $GL(n,\mathbb{C})$. Denote $Z(G)$ by the center of $G$. ${\bf Question}:$ Is it true that every finite normal subgroup of $G$ are ...
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Decomposition of Lie algebras using Weyl's Reducibility

Suppose I have a semisimple ideal $\mathfrak{g}$ of a Lie algebra $\mathfrak{l}$, is it possible to uniquely write $\mathfrak{l}=\mathfrak{g}\oplus\mathfrak{i}$ where $\mathfrak{i}\subset\mathfrak{l}$ ...
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An Alternative Definition of Reductive Lie Algebra?

I came across an alternative definition of reductive Lie algebra as follows: $\mathfrak{g}$ is said to be reductive of all abelian ideals of it are contained in its center $Z(\mathfrak{g})$ and ...
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31 views

Finitely Generated Matrix Group Decompositions

If I take a finite collection of n x n invertible matrices and generate a group G under matrix multiplication, is it the case that there always exists a maximal normal solvable group R from which I ...
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1answer
55 views

Ideals of solvable lie algebra

Let us say that S is a Lie algebra of dimension $n$, which is also solvable. Is it true that S contains an ideal of each dimension $d$ for $0 \leq d \leq n$? If so, how? Thanks for all the help.
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The commutator of a Lie algebra element with a Lie group element

Is there a way to evaluate the trace of generators of the Lie algebra and group elements? For example take $SO(N)$, with $\lbrace T^a\rbrace$ the set of generators, normalized such that ...
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Questions about the indivisible imaginary root in affine root system.

I am reading the paper. On page 5, $\delta$ is defined as the indivisible imaginary root in $\widehat{\Delta_+}$. $\Lambda_0 \in \widehat{\mathfrak{h}^*}$ is the unique element satisfying $\langle K, ...
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Why is 1/2+1/2 in the weight space for SO(5)

Let's consider $\mathfrak{so}(5)$ as the Lie algebra of $\mathrm{SO}(5)$, where the symmetric bilinear form is $x_1y_5+\cdots +y_1x_5$. Then the maximal torus is given by $$\left(\begin{array}{cccccc} ...
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Moment map and Hamiltonian

Take the manifold $M$ to be $M=\mathbb{R}^6=\mathbb{R}^3\times\mathbb{R}^3$ (hence $x\in M$ is given by $x=(p,q)$ with $p$ and $q$ three dimensional vectors) and take the possion bracket on $M$ given ...
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Exercise 11, chapter 2 in Lie Groups, Lie Algebras, and Representations: An Elementary Introduction

I am reading the book: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction by Brian C. Hall. I am stuck at the following exercise: exercise 11, chapter 2 . Can you help me? ...
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1answer
31 views

Lie algebras with different bases

I am interesting to know that if a finite dimensional Lie algebra $L$ has two bases $\beta_1$ and $\beta_2$, how can we compare the cardinal of two sets $\{(x,y)\in \beta_1\times ...
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39 views

Embedding so(n) in su(n)

Is there any way of embedding $\mathfrak{so}(n)$ into $\mathfrak{su}(n)$ for any $n$ other than picking the antisymmetric matrices of $\mathfrak{su}(n)$? I know that for small $n$ one can use ...
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Reducible Lie Algebra

I'm furthering my physics knowledge through a book called Lie Algebras in Particle Physics and am having trouble with one aspect of a problem. I believe because it's a question purely about ...