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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$24\otimes 24$ notation for representations of $SO(24)$

I was working through a set of string theory notes, and I came across the notation $24 \otimes 24$ to denote a reducible representation of $SO(24)$, but I am not familiar with the notation. I have ...
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0answers
20 views

Definitions of length function on a Weyl group

Let $\Phi$ be an irreducible root system and $W$ the Weyl group of $\Phi$. Denote by $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_l\}$ the corresponding base. Can anyone give me the standard definition ...
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1answer
33 views

Derived algebra and centre of a Lie algebra are preserved under isomorphism

I want to prove that the derived algebra $\mathfrak{g}' = [\mathfrak{g},\mathfrak{g}]$ and the centre, $Z(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ are preserved under isomorphism. So first I ...
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1answer
26 views

$i_1,i_2$ are ideals of Lie algebra $\mathfrak{g}$, is $[i_1,i_2]$ also an ideal?

If $i_1$ and $i_2$ are ideals of $\mathfrak{g}$, I want to show that $[i_1,i_2]$ is also an ideal of $\mathfrak{g}$. This is how I proceed: $[i_1,\mathfrak{g}]\subseteq ...
2
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1answer
31 views

Are all irreducible representations of solvable Lie algebras 1-dimensional?

Let $\mathfrak{g}$ be a solvable Lie algebra. By Lie's theorem, it is easy to see that any finite dimensional irreducible representation is 1 dimensional. Is it possible to remove the condition that ...
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23 views

The Taylor series for product of Lie group elements

Let $x$ and $y$ be two elements of a Lie group $G$. In chapter 2 of the text "Lie Groups and Lie Algebras I" by A. L. Onishchik, the author states that, if $\overline{x}$ and $\overline{y}$ denote the ...
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1answer
54 views

Why ad(H) has trace 0, if H has trace 0

I'm reading some notes on Lie Algebras, and right now we are constructing a complex, 3-dimensional, simple nonabelian Lie algebra. Suppose $\mathfrak{g}$ is a 3-dimensional sub-Lie algebra of ...
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1answer
33 views

Adjoint representation - A working example

I'm working on Lie Algebra at the moment and while everything I hear and read about it makes sense, I get stuck with some exercises (The question How to determine the matrix of adjoint representation ...
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1answer
114 views

Representation of an Abelian Lie algebra

I have to do this exercise: Given an abelian Lie algebra $\mathfrak{h}$ on an algebraically closed field $\mathbb{F}$. Let $ \varphi : \mathfrak{h} \to \text{End}_{\mathbb{F}}(V) $ be a ...
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26 views

rank of exterior derivative

As i am sitting here reading some lecture notes on lie algebras I found myself getting stock because of the word "rank". As I understand rank, it's just the dimension of the image of a linear map and ...
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1answer
21 views

about derivation for Leibniz algebras

A left Leibniz algebra over the field $k$ is a vector space over $k$ with a bilinear map $[~,~]:L\times L \mapsto L$ satisfying $$[a,[b,c]]=[[a,b],c]+[b,[a,c]].$$ A derivation of $L$ is $\alpha :L ...
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1answer
15 views

Inversion of Lie subspace in Cartan decomposition

Consider a matrix Lie group $G\subseteq GL$ with Lie algebra $\mathfrak{g}$. Using the Cartan decomposition, we can write $\mathfrak{g}$ as $(\mathfrak{k},\mathfrak{p})$. Let now $k\in\mathfrak{k}$ ...
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2answers
125 views

How do I derive this formula from gauge theory?

This is Exercise 3.4.14 in R. W. Sharpe's Differential Geometry. Suppose $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and $H$ is a Lie subgroup of $G$. Let $\theta$ be a ...
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0answers
12 views

Understanding Weights and Roots

I'm refering to this book called Semi-Simple lie algebras in Particle Phsics by Cahn for understanding weights and roots as given by our instructor. It has a definition of weights on Pg.33 which is ...
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0answers
42 views

matrix lie group

We know the set of one-parameter subgroup generators of a (closed) matrix lie group G (a subgroup of a general linear group) is closed under lie bracket and addition. In fact I see that in most ...
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0answers
19 views

Find the dimension of the real Lie algebra su(n)={A∈sln(C)|A+A∗ =0}, A∗ =A(conjugate transpose)

Find the dimension of the real Lie algebra su(n)={A∈sln(C)|A+A∗ =0}, A∗ =At. Also have to Show that the map A∗ =At. C⊗R su(n)→sln(C), z⊗A􏰀→zA su(n)={A∈sln(C)|A+A∗ =0}, is an isomorphism (of complex ...
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0answers
20 views

Find a stratification of a Carnot group

My problem is how to find the stratification in an Carnot group (see here pag. 3). Let's make an example. Let $\mathbb R^3$ be endowed with a composition law $*$ that makes $(\mathbb R^3, *)$ a Lie ...
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1answer
20 views

If $A(t)\in SO(3)$, why is $\dot{A}(t)A(t)^{-1}\in\mathfrak{so}(3)$?

I am going through some notes on geometric mechanics. In the first section we let $A(t)$ denote some product of Euler rotations: $$ A(t)= \left( \begin{array}{ccc} \cos (\psi ) \cos (\varphi )-\cos ...
0
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1answer
40 views

Derive the commutation relation

I'm having difficulty in deriving the commutation relation: [Eij, Ekl] = $\delta$jk Eil - $\delta$il Ekj Here Eij is a matrix with null entries everywhere at the i'th row and j'th column, where it ...
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0answers
24 views

Can a point have a nontrivial isometry group?

This question is extremely related to this other question. In fact, a positive answer here directly implies a positive answer there. However, since it is a mathematically different question I decided ...
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0answers
31 views

One parameter subgroups

I am trying to solve a image registration problem where my deformations are one parameter subgroups of diffeo. i.e., solution to the equation; $ \partial \varphi(x,t) = v(\varphi(x,t)). $ The ...
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1answer
25 views

On normal subgroups of $Iso(2)$

I want find all normal lie subgroups of $Iso(2)$. So i starting from finding normal subgroups of connected component of unity: $Iso^{\circ}(2)$, which corresponds to ideals of $\mathfrak{iso}(2)$. My ...
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0answers
34 views

Christoffel symbols and Riemann curvature tensor of a left-invariant metric on a Lie group

Let $G$ be a Lie group equipped with a left-invariant metric, with dimension n. One can write the local coordinates of $G$ as $\phi^a$, whereby $a=1,2..n$. From Milnor's 1976 paper "Curvatures of Left ...
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1answer
30 views

Find matrix of bilinear form on Lie algebra?

I have a bilinear form $$\sigma_V:L\times L\rightarrow \mathbb{k}$$ $$\sigma_V(x,y)=tr(\rho_V(x)\rho_V(y)) \forall x,y \in L$$ and am looking for the matrix of this form. I am in the algebra $$L= ...
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0answers
25 views

Continuous complex finite dimensional irreducible representation of $GL_n( \mathbb C )$

What are all the continous finite dimensional irreducible representation of $GL_n( \mathbb C )$? I tried the following since the continuous irreducible representations of $GL_n( \mathbb C )$ are in ...
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0answers
20 views

Two different definitions for Lie Algebras for closed subgroup of $GL_n(\mathbb R)$

Let $G$ be a closed subgroup of $GL_n(\mathbb R)$. There are two definitions for $\mathrm{Lie}(G)$ $\mathrm{Lie}(G) = \{ \gamma'(0) : \gamma : (-\epsilon, \epsilon) \rightarrow G \text{ is ...
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0answers
38 views

Are $h$-eigenspaces of infinite-dimensional $sl(2,\mathbb C)$-modules of dimension at most $1$?

Let $(\pi,V)$ be an irreducible representation of the Lie algebra $sl(2,\mathbb C)$ on a possibly infinite-dimensional complex vector space $V$. Further let $h,e^+, e^-$ be the usual standard basis ...
3
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0answers
23 views

Radical of an infinite dimensional Lie Algebra

I am studying Lie Algebras and I just encountered the notion of radical $R$ of a finite dimensional Lie algebra $L$ over a field $F$, the maximal solvable ideal. Since the dimension of $L$ is finite, ...
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0answers
22 views

Finite-Dimensional Representations of The Classical Groups in Tensor Spaces: Invariant Theory

I. When we study finite-dimensional irreducible representations over the space of general tensors (e.g.,Chapter 13, Group Theory in Physics by Wu-Ki Tung), is it enough to obtain all the ...
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1answer
19 views

An irreducible representation of a complex Lie algebra is the product of a 1-dim rep'n and a semisimple one

I am reading on p.128 of Fulton and Harris's Representation Theory the proof of the following fact about lie algebras Let $\mathfrak{g}$ be a complex Lie algebra, and set $\mathfrak{g}_{ss} := ...
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0answers
41 views

Exactness of the lie algebra functor

A question that occurs naturally but of which I (and, apparently, Google too) does not know an answer is the following: is the functor that associates to a lie Group its lie Algebra an exact functor? ...
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0answers
12 views

Relation between Yangian $Y(sl_2)$ and quantum affine algebra $U_q(\widehat{sl_2})$.

What is the relation between the definitions of Yangian $Y(sl_2)$ and quantum affine algebra $U_q(\widehat{sl_2})$? There are two definitions of $U_q(\widehat{sl_2})$. The following is Jimbo ...
2
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1answer
20 views

Relation between quantum affine algebra $U_q(\widehat{sl_2})$ and the affine Lie algebra $\widehat{sl_2}$?

The relation between the definition of quantum group and correpsonding lie algebra is discribed here. Are there some similar relation between $U_q(\widehat{sl_2})$ and $\widehat{sl_2}$? There are two ...
4
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43 views

No local optima in quantum control?

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation: $\frac{d x(t)}{dt} = F(x,u)$ one can consider the ...
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2answers
83 views

Is some One Dimensional subalgebra an Ideal of the 2 Dimensional Non-Abelian Lie Algebra?

Is there any one dimensional subalgebra which is an Ideal of the two dimensional non-abelian Lie Algebra? i.e. is it invariant as a subalgebra of the 2D non-abelian algebra I read that "all the ...
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0answers
45 views

Finding the tangent space at the identity *without* using paths

I know that there are two commonly used definitions of the tangent space at a point of a smooth manifold $M$: one using paths through that point, and the other using the idea of linear maps ...
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2answers
151 views

Advanced beginners textbook on Lie theory from a geometric viewpoint

There are several questions resembling this one but none of them are quite the same I believe. I have a background in differential geometry and topology, as well as analysis (locally convex spaces). ...
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2answers
55 views

Are Structure Constants of a Lie Algebra always Totally Antisymmetric?

Are the Structure Constants $c^a_{bc}$ of a Lie Algebra always totally antisymmetric so, $$ c_{abc} = c_{bca} = c_{cab} $$ Or is this just the case for semi-simple algebras?
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1answer
38 views

Tannakian category

Could you please explain to me what is meant by the notation : $ \mathrm{ad} $ that appears on the following link : www.jmilne.org/math/xnotes/tc.pdf in the middle of the page : $ 36 $. The level ...
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1answer
25 views

Reflection transforms positive root into a positive root.

I am reading the book "Simple groups of Lie type" by R.Carter, and stuck with the following lemma: Let $r \in \Pi$. Then $w_r$ transforms $r$ into $-r$ but every other positive root into a positive ...
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1answer
27 views

Proving a 1 Dimensional Lie Algebra is Abelian

This question is pretty basic but I am but a simple physics postgrad studying Lie Algebras for the first time and want to check my understanding... I am interested in showing that a rank one lie ...
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2answers
41 views

Tensor product of irreducible representations

Let $\mathfrak{g}$ be a complex simple finite dimensional Lie algebra and $V,W$ two irreducible finite dimensional representations. When is $V\otimes W$ irreducible?
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1answer
32 views

Fundamental representations of lie algebras

All the lie algebra considered are over $\mathbf{C}$. I know that for the lie algebra $\mathfrak{sl}_{n+1}$ the fundamental representations $L(\omega_k), k \in \{1,\cdots,n\}$ are the $\Lambda^k V$ ...
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1answer
62 views

Why are there no nonabelian Lie groups with dimension two?

I've been asked this immediately after been asked to show that the structure constants $c_{ijk}$ are totally antisymmetric, so I suppose there must be a connection, although I can't figure out where. ...
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0answers
7 views

Are Whittaker functions for lie algebras the same as Whittaker functions for corresponding Lie groups?

Some papers call some Whittaker function the Whittker function for some Lie group $G$. Some other papers call some Whittaker function the Whittker function for some Lie algebra $g$. Is the Whittaker ...
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0answers
21 views

Representation of a Kac-Moody algebra

Let $n$ be an integer $\geq 3$ and let $\mathfrak{g}$ be the Kac-Moody algebra with cartan matrix $C$ given by $C_{ij} = 2 \delta_{i,j} - \delta_{i,j+1} - \delta_{i,j-1} - \delta_{|i-j|,n-2}$. For ...
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0answers
30 views

How does a Lie algebra act on a tensor product of L-modules?

What is the $L$-module structure on $V\otimes W$ where $L$ is a Lie-algebra and $V,W$ are $L$-modules? The following question is related but I can't find the definition in there: tensor product of ...
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1answer
79 views

Generators of a semi simple lie algebra must be traceless

Consider a semi simple lie algebra. Show that if $T_a$ are the generators of a semi simple Lie algebra then $\text{Tr}T_a=0$. Attempt: $[T_a, T_b] = ic^c_{ab}T_c \Rightarrow \text{Tr}[T_a, T_b] = ...
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1answer
46 views

How are groups with the same Lie Algebra inequivalent?

I thought that groups with the same Lie Algebra are automatically equivalent, but there appear to be some exceptions to this? What sort of exceptions are there and why?
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23 views

How is a Sim3 lie group used to project a 3D point?

I have a $Sim3$ transformation with scale, rotation and translation that describe the camera to world transformation. How can I use it to transform a point in world coordinates to camera coordinates? ...