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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Induced actions.

Let $U$ be the subgroup of $GL_n$ consisting of all unipotent upper triangular matrices. Suppose that there is an action of $U$ on a variety $X$. Let $\mathbb{C}[U]$ be the group algebra of $U$ and ...
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Characters of a unipotent group.

Let $\def\C{\mathbb{C}}T = (\C^*)^n$. A character of $T$ is defined to be a homomorphism from $T$ to $\C^*$. The characters of $T$ is of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some ...
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Why is U(n) a real form of GL(n)

When $n=1$, we see that $U(1)$ is defined by the equation $z\bar z=1$, hence $a^2+b^2=1$ for $z=a+bi$. Taking complex $a,b$ we see that the solutions are nonzero complex points, hence $U(1)$ is ...
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25 views

Liesubgroups given by representations

Maybe it's a little odd question, but I encountered a classification of homogeneous spaces, and I'm stuck with the following description (c.f. Onishchick, "Topology of transitive transformation ...
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134 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 $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
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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 ...
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49 views

Does the set of $n$ by $n$ matrices of rank $q$ form a manifold?

I'm not sure whether the space of all rank-$q$ square matrices of dimension $n$ is a submanifold. I have totally no clue. Can somebody help?
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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 ...
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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 ...
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10 views

Local Lie Groups

Hello i am trying to prove the following proposition : Let $G$ be a connected Lie group, and $U\subset G$ a neighborhood of the identity element. Also, let $U^k = \{g_1 . g_2 . \dots g_k : g_i \in ...
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1answer
85 views

What does it mean to be a real Lie group

What does it mean to be a real Lie-group ? For example it is said that $SU(N)$ is a real Lie-group. While for example for $SU(2)$ the 2 dimensional matrix-representation consists of the Pauli ...
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39 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 ...
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1answer
62 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 ...
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23 views

Decomposition of direct sum representation of a Lie Group

Suppose that $(\varphi,V)$ is an irreducible finite dimensional representation of a Lie group $G$ and let $\psi=\bigoplus_{i=1}^n{\varphi}$ the representation on $W=\bigoplus_{i=1}^n{V}$. I want to ...
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18 views

If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler > structure?

A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which acts on $M$ and preserves complex structure then If $M$ has Kaehler manifold, then the symplectic quotient of $M$, i.e, ...
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1answer
41 views

Equivalence between vector field and generator of a group of translations

I've been reading Olver's Applications of Lie groups to differential equations and did not understand the excerp where the autor explains that "[...] every nonvanishing vector field is locally ...
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69 views

When do invariant measures arise from smooth differntial forms?

It is well known that the Haar measure of a Lie group $ G $ arises from a invariant differential form density $ |\omega| $ (of top dimension). Also, we know that if we have a closed subgroup $ H \leq ...
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94 views

Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...
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2answers
53 views

Introduction to discrete subgroups of the euclidean group

I am looking for a general introduction to discrete subgroups of the euclidean group (= group of isometries in euclidean space). Even though I searched quite a bit, I was unable to find a good ...
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1answer
35 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 ...
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48 views

Continuous subgroup of SO(3)?

I read from a paperarXiv: cond-mat/0602109 by a theoretical physicist, Prof. Frank Bais, close subgroups of $SO(3)$ is given by ${C_n,D_n,T,O,I,SO(2)\rtimes Z_2}$, where $C_n$ is the cyclic group of ...
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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?
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60 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 ...
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33 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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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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30 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 ...
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1answer
38 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 ...
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1answer
61 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
61 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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32 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, ...
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2answers
70 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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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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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
58 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} ...
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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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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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21 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
76 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
47 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 ...
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48 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 ...
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1answer
58 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 ...
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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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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
56 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
139 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? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
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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 ...
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0answers
13 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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45 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 ...