A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group. Consider using with the ...

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Assign a root to a irreducible representation

Given a root, e.g. $(-1 0 1 00)$ of $\text{SO}(10)$, how can I see/find to which representation of the Lie group it belongs?
4
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1answer
61 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
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34 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 ...
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1answer
26 views

Question on $\mathrm{Ad}(SL(3,\mathbb R))$

The following question appears in an example in page 116, Representation theory of semisimple groups, A. W. Knapp. Consider $G=\mathrm{Ad}(SL(3,\mathbb R))$. It is a subgroup of $GL(8,\mathbb C)$. ...
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1answer
34 views

Elements Outside the Identity Component $SO^+(1,\,3)$ of the Lorentz Group $O(1,\,3)$

I have been answering a question on Physics Stack exchange to do with the difference between the "proper orthochronous" (i.e. identity component of) the Lorentz group $SO^+(1,\,3)$ and the Lorentz ...
2
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1answer
40 views

Isomorphism $U(p,q)/U(1)=SU(p,q)/Z_{n}$

I have a little experience in Lie groups, so I have met the strange isomorphism: $$U(p,q)/U(1)=SU(p,q)/Z_{n}$$ Here $U(p,q)$ is a set of complex $n\times n$ matrices ($p+q=n$), which satifies the ...
1
vote
1answer
64 views

Isn't the picture on Wikipedia about Weyl Chambers wrong?

Wikipedia's article on Weyl groups shows an example of a root system and the corresponding fundamental chambers (in my understanding, also known as fundamental regions or fundamental domains). In this ...
3
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1answer
62 views

What is the maximal torus in the Lorentz group $O(m,n)$?

I'm close to certain it's just the product of the maximal tori of $O(m)$ and $O(n)$, but I can't quite prove it. I've tried the following: ...
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40 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
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2answers
72 views

Homogeneous space question: a quotient $U(n)/U(n-1)$

One can block-diagonally embed a copy $H$ of the unitary group $U(n-1)$ into $U(n)$ by $$A \mapsto \begin{bmatrix}\det(A)^{-1}&0\\0& A\end{bmatrix}.$$ According to a remark in the example ...
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1answer
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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, ...
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2answers
77 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 $SO(3)$, ...
5
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1answer
39 views

Commuting path from identity to matrix

Let $G$ be a connected, closed subgroup of $\operatorname{GL}(n,\mathbb{C})$ and let $g \in G$. Is there a continuous function $f:[0,1] \to G$ such that $f(0) = g$ and $f(1)=1$ and $f(t) \cdot g = g ...
2
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0answers
62 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} ...
1
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1answer
24 views

Covolution (space) over compact Lie groups

Let $G$ be a compact Lie group. Is there any way one can characterize the functions $\phi$ of the form $\phi=\psi\ast \psi^\ast$ in $C^\infty(G)$ where $\psi\in C^\infty(G)$? Here as usual ...
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16 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 ...
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22 views

Do the infinitesimal generators of a group need be necessarily exponential

A vector field is supposed to be an infinitesimal generator of a lie subgroup. Typically to generate a flow on a manifold using a one parameter group of transformations, we do exponentiation of the ...
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1answer
85 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
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2answers
53 views

Equivalent form for the Bruhat decomposition

Let $G$ be a reductive group and $B$ a Borel subgroup. The Bruhat decomposition allows us to write (where $W$ is the Weyl group): $$ G/B = \coprod_{w\in W} BwB$$ Why is this form the same as looking ...
3
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0answers
49 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
1answer
62 views

Group representations and short exact sequences

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequence of groups. What can be said about group representations of $B$ if we assume a complete classification of the ...
1
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1answer
29 views

Conditions for a matrix in $Bw_0B$.

Let $B$ be the set of all upper triangular matrices in $GL_n$. What are the conditions for a matrix in $GL_n$ lies in $Bw_0B$ (What do the matrices in $Bw_0B$ look like)? Thank you very much.
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46 views

Relationship between O(n)- and SO(n)-representations?

Write $O(n)$ and $SO(n)$ for the orthogonal and special orthogonal group of degree $n$ over the real numbers. Suppose that $V$ and $W$ are real, finite-dimensional and orthogonal ...
3
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1answer
63 views

tangent/normal space to set of symmetric isospectral matrices

Let $\Lambda = \{\lambda_1, \ldots, \lambda_n\}$ be a set of $n$ distinct real numbers. $M_n(\mathbb{R})$ denotes the set of all $n \times n$ real matrices, and for $B\in M_n(\mathbb{R})$, $B^T$ ...
6
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1answer
145 views

Confluent Heun equation. Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
2
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0answers
36 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 ...
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14 views

Reference request: Tensor Product of $m$ $SU(N)$ algebras

I'm working on quantum mechanics of linear $SU(N)$ chain of sites. Specifically, I would like to study a Tensor product of $m$ Lie algebras $\mathfrak{s}\mathfrak{u}(N)$ $$ V\in SU(N)\\ W = ...
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26 views

Question about a notation.

I have a question about the notation in the paper. On page 8, the 3-rd line from bottom, it is said that $h_J(t)$ is the corresponding diagonal matrix in $GL_4$. I think that $J$ is some set such that ...
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46 views

Questions about unipotent matrices.

I have a question about the notation in the paper. On page 8, the 5-th line from bottom, it is said that $u_J(t)$ denotes the upper triangular unipotent matrix with $t$ in position corresponding to ...
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0answers
41 views

Lie group representatiom - quasi-equivalent representations

Let $T$ and $U$ be unitary representations of a conected simply conected nilpotent Lie group, such that all irreducible subrepresentations of $T$ and $U$ are the same. If $T$ and $U$ are finite, then ...
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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 ...
3
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1answer
70 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 ...
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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:= ...
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2answers
50 views

How to prove this property of a projective transformation?

The copy below is from this book: Sophus Lie, Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen, bearbeitet und herausgegeben von Dr. Georg Wilhelm ...
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0answers
33 views

Is there a name for the function $g(z)=\frac{1-e^{-z}}{z}$?

This function comes in connection with exponentiation for matrix Lie groups, e.g. in computing the derivative of $\exp()$ away from the identity, or $\exp()$ of something in the affine group ...
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18 views

$U(n)/Z(U(n))$ is isomorphic to $SU(n)/Z(SU(n))$

A problem that I have been working is withere $U(n)/Z(U(n))$ is isomorphic to $SU(n)/Z(SU(n))$. I believe the best way to approach this problem is to show that they are both Isomorphic to the same ...
1
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1answer
55 views

Decompose complex vector by SU(4)

This question is about to decompose (or reduce dimension) complex vector by $SU\left( 4 \right)$. Given any $4\times1$ complex vector $B$. We can build $a_i$,and matrix$\lambda_i,i=1\ldots n $, $a_i ...
0
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1answer
25 views

$T_1 \times T_2$ is a maximal torus?

I have been working on teaching myself matrix groups and I have come across a problem about maximal tori. If I have a torus, $T_1 \subset G_1 $ and it is the maximal torus and If I have a torus, $T_2 ...
1
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1answer
49 views

prove that the maximal torus of $SO(3)$ is the maximal torus of $GL_3(\mathbb{R})$.

I want to prove that the maximal torus of $SO(3)$ is the maximal torus of $GL_3(\mathbb{R})$. I want to use the theorem that every maximal torus of G equals $gTg^{-1}$ for some $g \in G$. But I am not ...
5
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1answer
41 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 = ...
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0answers
59 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
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28 views

Unimodular groups

Let F be a non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq (where q is a power of p). Is GL(n,F) ...
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0answers
34 views

Three linearly independent vector fields

How can one find three linearly independent vector fields on $S^1\times S^2$? I know that $S^1\times S^2 \cong SO_3( \mathbb{R})$, i.e. the set of orthogonal $3 \times 3$ matrices with determinant ...
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0answers
41 views

$U(n)$ and $U(1)\times SU(n)$ are not isomorphic Lie groups

I am reading John Lee and on the chapter about group actions there is a problem that asks me to show that $U(n)$ and $U(1)\times SU(n)$ are not isomorphic Lie groups by showing that they don't have ...
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79 views

How can we extend the proof that Lie groups are orientable to parallelizable manifolds?

Let $G$ be a Lie group with identity element $e$ and dimension $n$. I know we can take a non-vanishing $n$-form (call it $\omega$) on the vector space $T_eG$. For a Lie group, as we have an ...
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2answers
55 views

Lie bracket of vector fields on $R^2$

Compute the Lie bracket$$\Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big]$$ on $R^2$ Can you help me please?
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15 views

About representations and transformations under an $SU(n)$ Lie Group

I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the ...
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1answer
36 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 ...
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29 views

Why is the Compact Symplectic Group Simply Connected

Let $Sp(n)=U(n,\mathbb{H})=\{A \in M_{n}(\mathbb{H}) : A\cdot A^{*}=I\}$ be the compact symplectic group, a subset of $Sp(2n,\mathbb{C})$. I want to show that $Sp(n)$ is simply connected, in ...
2
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1answer
58 views

Question about vector fields and Lie group

Notation: $\chi(G)$ is the set of smooth vector fields on Lie group $G$, which in fact forms a vector space. Given a Lie group $G$, show that there exists a smooth vector field $X\in \chi(G)$, ...