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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$[L_+^m, L_y^n]$ in the $SO(3)$ Lie Algebra

Let $SO(3)$ be generated by infinitesimal rotations $L_x, L_y, L_z$ such the typical relations $ [L_x, L_y] = L_z $ and similar. Let $L_\pm = L_x \pm i L_y$ be the raising and lowering operators. Is ...
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191 views

lie group and lie algebra: confusion about the tangent space

I am reading a document about performing 2D and 3D transformations in space specifically rigid body transformations which can be represented using Lie groups. For example, the rigid body ...
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1answer
41 views

Confused about classification of simple lie algebras

In the classification of simple lie algebra, I learnt that $\mathfrak{sl}(n + 1)$ has a root system $A_n$. For example http://stacky.net/files/written/LieGroups/LieGroups.pdf, page 82. But I found ...
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1answer
37 views

Natural Lie algebra representation on function space

There is natural Lie group representation of $GL(n)$ on $C^\infty(\mathbb{R}^n)$ given by \begin{align} \rho: GL(n) & \rightarrow \text{End}(C^\infty(\mathbb{R}^n)) \\ A & \rightarrow \left(...
2
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1answer
96 views

Tensor product over Lie group isomorphic to that over its Lie algebra

Hi: I have a question about the tensor product of representation of Lie groups as follows: Let $G$ be a connected, complex Lie group with Lie algebra $\mathfrak{g}$. Let $V$ and $W$ are ...
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1answer
38 views

Notation in Kac Problem 3.2

I'm working through Kac's book, "Infinite Dimensional Lie Algebras", and have come across some notation I find confusing. Here, $e$, $f$, and $h$ are the Cartan generators of $\mathfrak{sl}_{2}(\...
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56 views

Irreducible Representations of Nilpotent Lie algebras

By Lie's theorem all irreducible representations of a solvable Lie algebra over $\mathbb C$ are one dimensional. What are all irreducible representations of a nilpotent Lie algebra ?
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363 views

Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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64 views

Does the adjoint action induce a trivial action on Lie algebra cohomology?

Let $k$ be an algebraically closed field and $\Gamma$ a finite group. $\Gamma$ acts on itself via conjugation, and it is true that the induced action on the cohomology algebra $H^{*}(\Gamma,k)$ is ...
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1answer
122 views

Weyl groups: correspondence of reflections and roots?

If W is the Weyl group of some ADE-type Lie algebra, and w is an element corresponding to a reflection (not just an involution), does it necessarily correspond to a root?
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89 views

Inductively prove that $L_{\mathbb{X}}i_\mathbb{Y}=i_{[\mathbb{X},\mathbb{Y}]}+i_\mathbb{Y}L_{\mathbb{X}} $

Let $\mathbb{X}$, $\mathbb{Y}$ be vector fields on $U \subset \mathbb{R}^n$. Prove that $$L_{\mathbb{X}}i_\mathbb{Y}=i_{[\mathbb{X},\mathbb{Y}]}+i_\mathbb{Y}L_{\mathbb{X}} $$ using induction. Assume ...
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1answer
238 views

Why are root spaces of root decomposition of semisimple Lie algebra 1 dimensional?

I'm trying to understand root system of semisimple Lie algebra but having trouble following one of the step in the note which explain why each root spaces are 1-dimensional. According to the note, ...
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28 views

Homologie of Lie algebra with coefficients in tensor product of modules

I'd like to prove that $H_i(\mathfrak g, M\otimes N)=\operatorname{Tor}_i^{U \mathfrak g}(M,N).$ My idea was that $H_i(\mathfrak g, M\otimes N)=\operatorname{Tor}_i^{U \mathfrak g}(k,M\otimes N)$. ...
3
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1answer
143 views

Algebra of matrix coefficient over a compact group is isomorphic to its dual.

Let $K$ be a compact group. Then we have the following definition of matrix coefficient: Definition: $f: K \rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite dimensional ...
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1answer
55 views

Symmetrizability of generalised Cartan matrix

How to prove that a generalized Cartan matrix whose diagram contains no cycles is symmetrizable? Any hint would be sufficient. Thanks in Advance.
3
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1answer
56 views

How to find coefficients in Lie bracket relations in Cartan-Weyl basis?

For example, consider an $D_n$ Lie algebra. The Cartan-Weyl basis satisfies the following Lie bracket relations [1, p.98] $$ \left[H^{\alpha},H^{\beta}\right]=0 $$ $$ \left[H^{\alpha},E^{\beta}\...
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2answers
140 views

Prove an identity using differential calculus to a problem connected to fluids

Euler's equation for a incompressible inviscid fluid is $\displaystyle \frac{\partial \textbf{v}_t}{\partial t}+(\textbf{v}_t \cdot \nabla)\textbf{v}_t=-\nabla p_t$ where $\textbf{v}_t=\...
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38 views

What is $\Lambda^{i}$ in the “Show that the highest weight of $\Lambda^{i}V $ is $\omega_{i}”$?

Question: What is $\Lambda^{i}$ in the "Show that the highest weight of $\Lambda^{i}V $ is $\omega_{i}$"? In this question, $\omega_{i}$ are fundamental weights. Context: Highest weight modules of ...
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1answer
85 views

How to show trace of $AB$ is zero for $A \in \mathfrak{u}_n$ and $B \in \mathcal{H}_n$?

Please have a look at this question: Help needed in understanding the basics of Cartan decomposition of a Lie algebra I want to show that the decomposition $\mathfrak{gl}_n = u_n \oplus \mathcal{H}...
3
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1answer
221 views

Help needed in understanding the basics of Cartan decomposition of a Lie algebra

I am trying to learn the basics of Cartan decomposition of Lie algebra, and have come across the following example. Consider $\mathfrak{gl_n}$ as the Lie algebra of endomorphisms of $\mathbb{C}^n$. ...
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1answer
63 views

simple lie algebras

Is there any way of directly proving that the lie algebra $\mathfrak{sl}(n,\Bbb C)$ is simple? I am not asking for a complete proof, but could somebody please give me a hint on how I can proceed?
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1answer
35 views

Dimension of image of Lie bracket

Is there a method to calculate the dimension of the set of vectors in $\mathfrak{su}(n)$ $\{\ [A,B] \ \text{s.t} \ B \in \mathfrak{su}(n)\}$ for some fixed $A$. Is the dimension the same for all $A$?
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75 views

Relation between simple roots and fundamental weights.

Let $\alpha_1, \ldots, \alpha_n$ be simple roots of a semisimple complex Lie algebra. Let $\omega_1, \ldots, \omega_n$ be the fundamental weights. We have $$ \alpha_i = \sum_{s} k_s \omega_s, $$ for ...
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28 views

Does an arbitrary product of $f$ and $f^\dagger$ belong to a universal enveloping algebra of the Heisenberg algebra?

The Heisenberg algebra is essentially the canonical commutation relations (CCR) for bosons $[f,f^\dagger]=1$. $f$ is called an annihilation operator in physics ($f^\dagger$ creation operator). ...
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143 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 ...
2
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1answer
78 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 $\pi_{m,\,n}: \text{SU}(n) \to \text{GL}(V_m)$....
3
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1answer
44 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 $-1$...
3
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2answers
215 views

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

Let $\mathbb{X},\mathbb{Y}$ be vector fields on $\mathbb{R}^n$. Let $\Phi_t$ denote the flow of $\mathbb{X}$. Define $L_{\mathbb{X}}\mathbb{Y}=[\mathbb{X},\mathbb{Y}]$. You are given that $\...
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1answer
39 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 $F^*(L_{\mathbb{Y}}\omega)=L_{\...
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1answer
121 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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2answers
35 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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199 views

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 ...
3
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1answer
61 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
48 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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1answer
386 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?
2
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1answer
50 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 ...
3
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1answer
30 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 $e,...
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33 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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0answers
276 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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62 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 ...
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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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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
201 views

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

Is the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ simple? Wikipedia states that the Lie algebra $\mathfrak{sp}(2n,\mathbb{R})$ is simple. http://en.wikipedia.org/wiki/Table_of_Lie_groups ...
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1answer
215 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? ...
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1answer
54 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
67 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, we ...
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1answer
88 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
80 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 ...
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2answers
51 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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87 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 ...