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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Why is this generator an element of the Cartan subalgebra of SU(3)?

In the solution of Problem 3 of these notes (papge 4), it stated that $\lambda_2$, $\lambda_5$ and $\lambda_7$ form a SU(2) subalgebra of SU(8), where $\lambda_i$ are the Gell-Mann matrices. In ...
3
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34 views

Are there common inequivalent definitions of Cartan subalgebra of a real Lie algebra?

I'm confused about seemingly different notions of a Cartan subalgebra of a real semisimple Lie algebra, and I'm wondering if there are common inequivalent definitions. In the book Lie Groups: Beyond ...
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1answer
42 views

representation $\pi_{m,\,n}: \text{SU}(n) \to \text{GL}(V_m)$

Let $V_{m,\,n}$ denote the vector space of the homogeneous complex polynomials of degree $m$ in $n$ variables (under addition). Define a representation $\pi_{m,\,n}: \text{SU}(n) \to ...
3
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1answer
27 views

Character of a tensor product of $\mathfrak{sl}_2$-modules

Let $V$ be a finite-dimensional $\mathfrak{sl}_2$-module. There is a standard base $\{e,f,h\}$ in $\mathfrak{sl}_2$, I use standard notation ($h$, for instance, is the diagonal matrix with $1$ and ...
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1answer
65 views

Proving $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$

Let $X,Y$ be vector fields. $L_X$ is the Lie derivative and $i_X$ is the contraction of a $k$-form. I am really stuck on how you could prove the identity $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$. Update: I ...
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28 views

Proving $L_{\mathbb{X}}i_{\mathbb{Y}}=i_{[\mathbb{X},\mathbb{Y}]}+i_{\mathbb{Y}}L_{\mathbb{X}}$ [duplicate]

Let $\mathbb{X}$, $\mathbb{Y}$ denote vector fields on $U \subset \mathbb{R}^n$. Prove the identity $L_{\mathbb{X}}i_{\mathbb{Y}}=i_{[\mathbb{X},\mathbb{Y}]}+i_{\mathbb{Y}}L_{\mathbb{X}}$ I ...
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12 views

Proving that $\Phi_{t*}[\mathbb{Y},\mathbb{Z}]=[\Phi_{t*}\mathbb{Y},\Phi_{t*}\mathbb{Z}]$

Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$. Let $\Phi_t$ denote the flow of $\mathbb{X}$. You are given that $\displaystyle L^j_{\mathbb{X}}[\mathbb{Y},\mathbb{Z}]=\sum_{k=0}^{j} ...
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1answer
22 views

Showing that $F^*(L_{\mathbb{Y}}\omega)=L_{\mathbb{X}}(F^*\omega)$

Let $F:U \rightarrow V$ be a diffeomorphism between open sets in $\mathbb{R}^n$. Let $\mathbb{Y}$ be a vector field on $V$ and $\omega$ a $k$-form on $V$. Show that ...
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1answer
23 views

Why is the commutator subalgebra of a Lie group a linear subspace?

One defines commutator subalgebra of Lie algebra $\mathfrak{g}$ as $[\mathfrak{g},\mathfrak{g}]$. Why is it really subalgebra: Why $\forall_{a,b,c,d \in \mathfrak{g}} \exists_{e,f \in \mathfrak{g}} ...
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25 views

An exponential map of a matrix computation

Suppose the $n\times n$ matrices $A$ and $M$ satisfy $AM+MA^{T}=0.$ Show by direct computation that the product $\mathrm{exp}(At)~M~\mathrm{exp}(A^{T}t)=M$ for all $t\in \mathbb{R}.$ Note: By ...
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119 views
+50

Why the Steinberg idempotent is idempotent?

Consider the group $GL_n(\mathbb{F}_p)$. We have the following subgroups : -$\Sigma_n$ the symmetric group (permutation matrices) -$B_n$ the Borel subgroup (upper triangular matrices) -$U_n$ the ...
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25 views

What is an example of a map not satisfying this rank condition?

Definition: Consider a Lie Group $G$ and a set of right invariant vector fields on $G$, denoted $\Gamma$. A point $y \in G$ is called normally accessible from a point $x \in G$ by $\Gamma$ if there ...
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1answer
11 views

Definition of the Dynkin Diagram (in Humphreys)

I'm reading paragraph 11 in Humphreys' 'Introduction to Lie Algebras and Representation Theory'. The author defines Coxeter graphs and Dynkin diagrams for any rank-many distinct positive roots. He ...
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22 views

Why are lie algebra of upper-triangular $nxn$ matrices not nilpotent Lie algebra

Is there an easy proof (without Engel's theorem) of the fact that lie algebra of upper-triangular $n\times n$ matrices (of the field $\mathbb{R}$) are not nilpotent Lie algebra?
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1answer
21 views

Subalgebra condition in Engel's theorem

An equivalent version of Engel's theorem says that Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V\ne 0$, then there exists ...
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1answer
26 views

Lie subalgebra in $Der(\mathbb{C}[z])$ isomorphic to $\mathfrak{sl}_2$

I am to prove that $\{(az^2+bz+c)\frac{\partial}{\partial z}:a,b,c\in\mathbb{C}\}$ regarded as a Lie algebra is isomorphic to $\mathfrak{sl}_2(\mathbb{C})$. I guess it is possible to build a basis ...
2
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16 views

if X \in gl(V) is any nilpotent element, then the adjoint action ad(X) is nilpotent

I found a observation in the beginning of the proof of Engel's Theorem in the Fulton's book "Representation Theory" Observation: if $X \in \mathfrak{gl}(V)$ is any nilpotent element, then the action ...
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14 views

What is the Jacobian for Sim(3) lie group action on 3D points ? (4d homogenous points)

I am coding up Sim(3) constraint types for a factor graph, and need to calculate the jacobian of the Sim(3) group action on 3D points. I am following the guide on http://ethaneade.com/lie.pdf ...
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15 views

Decomposition of SU(n) anticommutator

In $SU(N)$, the special unitary group, the algebra generators $T_a$ are hermitian and traceless. The structure constants are fixed with $[T_a,T_b]=i f_{abc}T_c$. In the fundamental representation of ...
2
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0answers
21 views

Closure relations of the cells in the Bruhat decomposition of the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$. How do we prove the closure relations between the cells, which ...
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15 views

question about infinitesimal transformations

Lawrence Dresner says this: (p. 10, Applications of Lie's Theory of Ordinary and Partial Differential Equations) Assume you have two infinitesimal group transformations: $$x'=x+\varepsilon(\lambda - ...
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1answer
34 views

Is the symplectic group $Sp(2n,\mathbb{R})$ simple?

Is the symplectic group $Sp(2n,\mathbb{R})$ simple? Wikipedia states that the Lie algebra $sp(2n,\mathbb{R})$ is simple. http://en.wikipedia.org/wiki/Table_of_Lie_groups However it only lists ...
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1answer
36 views

Is the exponential map for $\text{Sp}(2n,{\mathbb R})$ surjective?

For $\mathfrak{g} := {\mathfrak s}{\mathfrak p}(2n,\mathbb{R})$ and $G = \text{Sp}(2n,{\mathbb R})$, is the exponential map \begin{equation} \text{exp} : \mathfrak{g} \to G \end{equation} surjective? ...
4
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1answer
24 views

Why does the maximal compact subgroup of a Lie group inject into the compact form?

I've seen multiple sources state the following (without proof or reference), but I don't see why it's true. Let $G$ be a Lie group, and $G_u$ be a compact connected Lie group such that the ...
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1answer
33 views

Lie algebra representations and tensor product decompositions.

Find the weights for $V_{L_1 - 2L_3}$, where $L_1, L_2, L_3$ are the weights for the standard representation of $\mathfrak{sl}_3 \Bbb{C}$ on $V \cong \Bbb{C}^3$. In order to find these weights, ...
0
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0answers
15 views

how to compute the norm of a root from the Cartan matrix?

As far as I understand, the Cartan matrix is associated with a unique semi simple algebra. How can we compute the norm of a root $\alpha$ from it since its components are invariant under rescaling? ...
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1answer
49 views

Decomposition into irreducibles of representations of semisimple Lie groups.

Let $G$ be a connected semisimple Lie group and $\mathfrak{g}$ it's Lie algebra. Then $\mathfrak{g}$ is semisimple. Let $V$ be a finite dimensional representation of $G$. Viewing $V$ as a ...
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1answer
28 views

Tangent space of matrix group is a Lie subalgebra

In my lecture today, we were covering matrix groups and Lie algebras. My professor made the statement that given any matrix group $G$, the tangent space of the group at the identity $T_{e}G$ is a Lie ...
2
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2answers
32 views

Finding the tangent space of a subgroup

My professor set the following question and I have an answer, though would like someone with more experience to cast a critical eye over the details as I don't necessarily trust my result! Define the ...
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0answers
68 views

How is the periodic structure of SO(n) reflected to its lie algebra so(n)?

An element of $SO(n)$ represents an rotation so that it must have identity with $2\pi$-like additional rotation. On the other hand, the elements of lie algebra $so(n)$ construct an noncompact vector ...
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13 views

How can the generators of subalgebra $\mathfrak g^{\sigma}$ of $\sigma$-stable elements be expressed through generators of Lie algebra $\mathfrak g$?

Let $\mathfrak g$ be the semisimple Lie algebra of type $D_{4}$. Let $\sigma$ be the 3-rd order automorphism of $\mathfrak g$ induced by the triality of $D_{4}$: $$ ...
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1answer
19 views

How to find the multiplicity of weight in a Verma module?

In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots. How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the ...
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7 views

The proportionality constant between the Casimir and the identity.

By Schur's Lemma, in any irreducible representation of a Lie algebra, the Casimir operator $J$ is proportional to the identity $Id$. How can we see that $J=j(j+1)Id$ for some natural number $j$ and ...
2
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49 views

Does the five lemma hold true for Lie algebras?

According to wikipedia, the Five Lemma is true in Abelian categories. But the category of Lie algebras is not Abelian. Then is the Five Lemma still true for Lie Algebras?
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18 views

The Lie algebra of special linear group of degree 2 over the set of complex numbers

I tried to compute the Lie algebra of $SL(2,\Bbb C)$. I wrote the followings: $sl(2,\Bbb C)$={$X\in M(2,\Bbb C)$: exp $tX$ $\in SL(2,\Bbb C)$}={$X\in M(2,\Bbb C)$: $det$ exp $tX$ =$1$}={$X\in M(2,\Bbb ...
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17 views

Find a vector field $\mathbb{Y}$ satisfying $L_{\mathbb{X}}\mathbb{Y}=\mathbb{Z}$

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $\mathbb{X}=(1,y)$. Let $\mathbb{Z}$ be the vector field on $\mathbb{R}^2$ given by $\displaystyle \mathbb{Z}(x,y)= \bigg( ...
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25 views

Find the lie derivative of a particular integral

Let $\mathbb{X}=(1,y)$ be a vector field on $\mathbb{R}^2$. Let $\Phi_t$ be the flow of $\mathbb{X}$. The flow of $\mathbb{X}$ I have calculated to be $\Phi_t(x,y)=(x+t,ye^t)$ Given a function ...
2
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1answer
23 views

Taking complex conjugate of an element in $\mathfrak{su}(2)_\mathbb{C}$

I read from a book that the complexification of the Lie algebra $\mathfrak{su}(2)$, noted $\mathfrak{su}(2)_\mathbb{C}$, is in fact the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$, the reason being: ...
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6 views

Determining the middle term of an exact sequence of Lie algebras

This is related to my previous question here. Suppose that $A_i, B_i$ are Lie algebras with $A_i$ is a sub-Lie algebra of $B_i$, $i=1,2,3$. Suppose that we have the following commutative diagram where ...
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45 views

How to define an action of vector field on $C^{\infty}(M)$?

Let $M$ be a manifold. Let $\hat{X}$ be a vector field on $M$. How to define an action of $\hat{X}$ on $C^{\infty}(M)$? Thank you very much.
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1answer
52 views

Determining Lie algebras from commutative diagrams of exact sequences

Suppose that we have the following commutative diagram of graded Lie algebras $$\begin{array} A 0& {\longrightarrow} & C_n & {\longrightarrow} & A_{n+1} &{\longrightarrow} & ...
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1answer
17 views

The Weyl group and eigenspaces

Let $V$ be a representation of the Weyl group. For any reflection $\sigma_{\alpha}$ (where $\alpha$ is a root), we know that $V$ has two eigenspaces with eigenvalues $1$ and $-1$. The ...
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1answer
31 views

polynomial algebras and their coefficients in prime fields

According to the definition of the polynomial algebras $A(n)$ and $A(n,m)$ for $ n \in \mathbb {N} $ and $ m \in \mathbb {N}^n$, if $\mathbb{F}$ be field $GF(2)$ and $ X_1,...,X_n$ be $n$ pairwise ...
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0answers
15 views

Normal subgroup invariant under $\text{Ad}_g$

Denote by $G$ a Lie group with corresponding Lie algebra $\text{Lie}(G)$. There the three maps inner automorphism/conjugation: $\text{Int}_g = L_{g^{-1}} \circ R_g \in \text{Aut}(G)$, $\text{Ad}_g ...
2
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2answers
41 views

The universal enveloping algebra of free Lie algebra is the tensor algebra on the Free Abelian Group?

Let $A$ be a set of cardinality at least 2 and let $M_A$ be the free abelian group generated by $A$. Let $L(A)$ be the free Lie algebra generated by $A$. I am reading On Injective Homomorphisms For ...
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1answer
34 views

The action of a Lie algebra on a manifold is a Lie algebra homomorphism. How to show it?

By definition, the action of a Lie algebra $\mathfrak g$ on a manifold $M$ is a Lie algebra homomorphism, $\mathcal A: \mathfrak g\rightarrow\mathfrak X(M), \xi\mapsto\xi_M$ such that the action map ...
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0answers
9 views

$H^q(\mathfrak{g},K;V)$ is equal to $Ext_{\left(\mathfrak{g},K\right)}^q\left(\mathbb{C},V\right)$?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $K$ be a closed subgroup of $G$ with corresponding Lie subalgebra $\mathfrak{k}$. Let $V$ be a $\left(\mathfrak{g},K\right)$-module. Then, I ...
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2answers
73 views

What is the nilradical of $\mathfrak{gl}_n$?

I'm really embarrassed to ask but what is the nilradical of the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$, i.e. the set of ad-nilpotent elements of $\mathfrak{gl}_n(\mathbb{C}) = ...
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1answer
27 views

Is the center of a compact Lie algebra precisely the set of vectors on which the Killing form is zero?

Suppose a Lie algebra $\frak{g}$ has a killing form, $B$, which is negative semidefinite. Suppose $B(X,X)=0$ for some $X\in \frak{g}$. Is $X$ necessarily in the center of $\frak{g}$?
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27 views

What is a Serre presentation of a Lie algebra?

For example, as in: Give a Serre presentation of Lie algebra $\frak{g}$ of type $G_{2}$. Is it the presentation in terms of Chevalley generators, which satisfy Serre relations?