0
votes
2answers
69 views

Why is $\mathfrak{sl}(n)$ the algebra of traceless matrices?

I'm studying Lie algebras as purely algebraic objects, without much of a background in the differential geometry surrounding Lie groups. The definition of $\mathfrak{sl}(n)$ has been given to me as ...
1
vote
1answer
67 views

What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
3
votes
2answers
54 views

Prerequisites to “Applications of Lie Groups to Differential Equations”

I'm currently a 4th year student at a university. I've become close with a professor and we talked about the topic of lie groups in differential equations. He then offered to do a reading course with ...
2
votes
2answers
53 views

The Weyl group of A_3

Could someone please list all elements of the Weyl group of the root system $A_3$ in terms of simple reflections. In this case the Weyl group is $S_4$. Its strange that GAP failed to list all elements ...
3
votes
1answer
35 views

Question on unitary representation of non-compact simple Lie groups

The following is an exercise appearing page 148 in Knapp's book, representation theory of semisimple groups. Let $G$ be a connected linear non-compact Lie group with simple Lie algebra $\mathfrak g$. ...
3
votes
0answers
66 views

Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
1
vote
1answer
31 views

Adjoint Lie algebra homomorphism

I have a problem deriving the adjoint action $ad_X(Y)=XY-YX$ from the adjoint transformation of the group on the Lie algebra. Background: The adjoint action of the Lie algebra on itself is given by ...
0
votes
0answers
19 views

Question on inner product on space of representations of compact Lie groups

Let $K$ be a compact connected Lie group, wiewed as subgroup of unipotent matrices. Let $G=\mathfrak{k}^\mathbb C$ be the complexification with Lie algebra $\mathfrak{g}=\mathfrak{k}\oplus ...
0
votes
0answers
28 views

Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
0
votes
1answer
25 views

Induced Lie algebra homomorphism from Lie group homomorphism: left translation

A general result of Lie Theory is that every Lie group homomorphism $\Phi: G\rightarrow H$ induces a Lie algebra homomorphism $\phi: \frak{g} \rightarrow \frak{h}$. Which Lie algebra homomorphism ...
7
votes
1answer
133 views

Special conformal killing fields - solving for integral curves.

For each $b\in\mathbb R^d$, let a vector field $X_b:\mathbb R^d\to\mathbb R^d$ be defined as follows: \begin{align} X_b(x) = 2(b\cdot x)x - x^2 b, \end{align} where $x^2 = x\cdot x$. This is the ...
1
vote
0answers
82 views

Itzykson-Zuber integral over orthogonal groups

I would like to know is there a closed form expression for the following Itzykson-Zuber integral for the orthogonal case. $I=\int_{\mathcal{O}(p)} ...
1
vote
0answers
21 views

What are the root systems for the n-dimensional torus?

My question may seem silly at first, but currently I am not able to work out the question of finding all roots for the n-dimensional torus. At first, it seemed obvious to me that there are no roots at ...
2
votes
0answers
55 views

Question on $\mathfrak{sl}(2,\mathbb R)$

I am confused about some facts on $SL(2,\mathbb R)$. The Lie algebra of $SL(2,\mathbb R)$ is $\mathfrak{sl}(2,\mathbb R)$. However, the map $$ \exp:\mathfrak{sl}(2,\mathbb R)\ \rightarrow ...
2
votes
0answers
29 views

Weyl Group of Parabolic subgroups

Let $G=SL(n,\mathbb R)$ with Lie algebra $\mathfrak{g}=\mathfrak{sl}(n,\mathbb R)$. The classical minimal parabolic subgroup $B$ consists of the upper triangular matrices. The parabolic subgroups $P$ ...
0
votes
0answers
14 views

Character of a symmetric square

Let $V$ be a representation of $\mathfrak{sl}_2(\mathbb{C})$. As far as I am concerned a character of $V$ is a Laurent polynomial $\sum_{k\in\mathbb{Z}}d_k\cdot t^k$, where $d_k$ is the dimension of ...
0
votes
1answer
13 views

Representation of $\mathfrak{sl}_2(\mathbb{C})$ corresponding to Lie algebra representation

We have a representation $R$ of a Lie group $\mathrm{SL}_2(\mathbb{C})$ in the space of polynomials $\mathbb{C}[x,y]$ such that $R\begin{pmatrix} a & b \\ c & ...
0
votes
0answers
17 views

Questions about an action of $U(\mathfrak{g})$.

Let $\mathfrak{g}$ be a Lie algebra and $U(\mathfrak{g})$ its universal envoloping algebra. Let $G$ be the Lie group of $\mathfrak{g}$ and $U$, $B^{-}$ the upper unipotent subgroup and lower Borel ...
0
votes
0answers
21 views

What is $d\mu$ of $\mu:T^*\mathbb{C}^n\rightarrow \mathfrak{gl}_n^*$?

This is an elementary question. Let $\mu:T^*\mathbb{C}^n \longrightarrow \mathfrak{gl}_n^*$ be the moment map given by $(x,y)\mapsto xy$. Then concretely, what is the differential $d\mu$ of $\mu$? ...
0
votes
0answers
21 views

Real or complex representation

How can one know for a given algebra $\frak{g}$ if a specific representation is real or complex? For example if $\frak{g}=so(10)$ how can one know that the representation $\underline{16}$ is complex? ...
6
votes
2answers
165 views

Lie Groups/Lie algebras to algebraic groups

I am reading some lie groups/lie algebras on my own.. I am using Brian Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction I was checking for some other references on ...
3
votes
0answers
31 views

Compact Lie algebras and Lie groups

A simple or semisimple Lie algebra is said to be compact if the $\mathrm{Tr}\left \{ T^\mathrm{adj}_{a}, T^\mathrm{adj}_{b}\right \}$ is positive definite where $T^\mathrm{Adj}_{a}$ are the generators ...
3
votes
1answer
118 views

Which subgroup of $\mathrm{SL}(2,\mathbb{C})$ is this?

I am looking into sub-algebras of $\mathfrak{sl}_2(\mathbb{C})$ and the subgroups of $\mathrm{SL}(2,\mathbb{C})$ they generate. The basis of $\mathfrak{sl}_2(\mathbb{C})$ I am using consists of 3 ...
0
votes
0answers
11 views

How to prove that $U(\mathfrak{h})$ is isomorphic to $\mathcal{O}(\mathfrak{h}^*)$.

Let $\mathfrak{h}$ be a Cartan subalgebra of a Lie group $G$. It is said that $U(\mathfrak{h})$ is isomorphic to $\mathcal{O}(\mathfrak{h}^*)$. Here $\mathcal{O}(\mathfrak{h}^*)$ is the ring of ...
0
votes
1answer
22 views

What is the Lie group of $\mathfrak{h}$?

Let $\mathfrak{h}$ be a Cartan subalgebra of a Lie group $G$. What is the Lie group of $\mathfrak{h}$? By definition, the Lie group of $\mathfrak{h}$ consisting of elements of the form $e^{h}$, $h \in ...
2
votes
0answers
17 views

Linearly independent skew symmetric complex matrices having the least eigenvalues

Question: Let $A$, $B$ be two $5 \times 5$ (or $7 \times 7$) skew-symmetric complex matrices (i.e. $A^t = -A$), and suppose that $$ \forall t,s \in \mathbb{C}, \quad M(t,s):=(tA+sB)^*(tA+sB) \text{ ...
1
vote
0answers
31 views

Lie bracket as defining element for transformations

Why is it precisely the Lie bracket that encodes the information about a given transformation? A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given ...
1
vote
0answers
42 views

Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
1
vote
1answer
31 views

Lie group and generated ideals

I have this question in my textbook, and I can't seem to solve it on my own: Let $P \subset GL(n,\mathbb{C})$ be a subgroup as following: $P$ consists of all matrices in block ...
2
votes
0answers
17 views

Does “Spherical Symmetry” as defined in General Relativity imply a Foliation of Spheres?

In Carroll's "spacetime and geometry" he defines a spherical symmetrical spacetime as a spacetime $(M,g)$ for which there exists a Lie algebra homomorphism between the Lie algebra of a subset of the ...
1
vote
1answer
33 views

Lie groups. How to show that the group operations are smooth.

$N:=\{g\in GL(n,R) : g_{ij}=0 \forall j>i , g_{ii}=1 ∀i\}$. For this matrix group, how can we show that it is a Lie group? I am at the beginning of the subject of Lie groups so I can not ...
2
votes
1answer
35 views

Is $sp(4)$ a subalgebra of $su(5)$?

Is $sp(4)$ a subalgebra of $su(5)$? And how can I prove/disprove this? I know already that it cannot be a regular maximal subgroup of $su(5)$ since the Dynkin diagram (which has two roots of unequal ...
1
vote
1answer
61 views

Lie groups, maps and the Weyl group

If I have a map of simple Lie groups $H \to G$, do I get a map of Weyl groups $W_H \to W_G$? If $H$ is the semisimple component of a parabolic subgroup then we can clearly get this (see Ivan's answer ...
3
votes
1answer
34 views

A maximal subalgebra of $E_6$ !?

I'm puzzeled by the following sentence in one of Baez's posts: The Lie algebra $E_6$ has a subalgebra of maximal rank isomorphic to $\mathfrak{so}(10)\oplus \mathfrak{u}(1)$. However, I thought ...
4
votes
1answer
57 views

The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
2
votes
0answers
32 views

Where does the name “toral” come from?

Where does the name "toral" come from in "toral subalgebra"? I know a little (very little) Lie groups theory, so I guess it could be related to a Lie group whose Lie algebra is the toral one. Is ...
1
vote
0answers
29 views

Adjoint Representation of Lorentz Group

I'm thinking about the image under the adjoint representation $\mathrm{Ad}$ of the proper (identity connected component) Lorentz group $SO^+(1,3)$. Since this group has a trivial centre (it contains, ...
-2
votes
1answer
39 views

reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
5
votes
2answers
63 views

Representations of the Special Orthogonal Group in Three Dimensions.

This will perhaps be an unenlightening question, but here I go. Hopefully someone can varify my thoughts. $\\$ Considering Lie Group Theory and Representation Theory, for the case of the $SO(3)$, ...
2
votes
0answers
56 views

Gradient of a real-valued function on SO(3)

I have struggling with a problem of evaluating the gradient of a cost function on the Lie group of rotations: SO(3). The cost is the following: \begin{equation} ...
0
votes
0answers
15 views

kac moody algebra and pde

I study PDE via Lie groups method, I also very much into Lie theory, including the infinite dimensional version. Recently I come across some infinite dimensional Lie algebra so-called Kac Moody ...
1
vote
1answer
71 views

Lie group and Lie algebra automorphisms

Assume that $G$ is a connected Lie group and that $\alpha:G\rightarrow G$ is an automorphism of $G$. Furthermore let $\alpha_*:\mathfrak{g}\rightarrow\mathfrak{g}$ be the corresponding tangent map at ...
3
votes
0answers
43 views

Actions of Weyl group

I get a feeling what I am going to ask is very standard and classic, but I am not able to find any reference. Any answer or reference would be appreciated. Let us assume that $G$ is a simply ...
1
vote
0answers
29 views

set of positive roots made negative by a Weyl group element

If $w$ is a Weyl group element of a simple lie algebra and the reduced expression for $w$ is $s_{i_1}s_{i_2}...s_{i_k}$ what are the positive roots (in terms of reflections from reduced expressions of ...
0
votes
0answers
33 views

Root space of a Semi simple group an LVS?

A semi-simple Lie group has a Cartan Subalgebra ($H$) (CSA) -an LVS, Dual to this CSA LVS is root space($H^*$), which is set funtionals that map elements of CSA to real numbers and hence useful in ...
2
votes
1answer
54 views

Quaternions, Lie Groups and Lie Algebras. Steps to realize a paper. [closed]

I have to realize a paper about quaternions and Lie Groups and Lie Algebras. How can I realize the links between quaternions and Lie Groups & Algebras. Which books do you recommend me? First, I ...
0
votes
0answers
15 views

Dual basis to $e_{i+1}-e_{i} \in \ker ((1,1,…1)^\vee\in(\Bbb E^{n+1})^\vee)$

Studying the root system $A_n$ given by the simple roots $v_i:=e_{i+1}-e_i \in \Bbb E^{n+1}/\Bbb R(1,1,...,1)$ for $i = 1,...,n$, I came across the following dual basis: $v_i^\vee:= ...
5
votes
1answer
40 views

Can the equality $e^{-tY}Me^{tY} = e^{tX}M $ be shown by showing it only to 1st order? (Lie representations)

We have that A and B belong to different representations of the same Lie group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations. $$A = ...
0
votes
0answers
46 views

Ranks of matrix Lie groups and Lie algebra of SU(1,1), SO(2,1)

I was trying to find out by Googling, but had no luck. Am I right in thinking that for the Lie GROUPS: rank SL(n,R) = n, rank SO(n,R) = n (not sure about this one), rank SU(n,C) = n-1 and ...
1
vote
1answer
31 views

Obtaining representations of $G$ from $\mathrm{Lie}(G)$.

Suppose $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$, and $\tilde{G}$ is the unique connected, simply connected Lie group whose Lie algebra is $\mathfrak{g}$. Let $C$ be any discrete ...