1
vote
0answers
35 views

Finding a lie group structure on $\mathbb R^n\setminus\{0\}$

I want to find all maps $g: \mathbb R^n\setminus \{0\} \rightarrow GL_n(\mathbb R)$ which satisfy the properties $g$ is differentiable and injective $g(g(a)b) = g(a)g(b)$ for all $a,b\in\mathbb ...
0
votes
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 = ...
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 ...
2
votes
0answers
36 views

Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
0
votes
1answer
45 views

Reference for introductory Lie Groups

I am currently learning about Lie groups,So kindly suggest a reference for Lie groups, which contains lecture on Manifolds as well.
2
votes
1answer
37 views

$SO(n)$ is a deformation retract of $SO(n,\mathbb{C})$

Does anyone know how to prove the fact that $SO(n)$ is a deformation retract of $SO(n,\mathbb{C})$? Here $SO(n,\mathbb{C}):= \{ A\in M_{n\times n}(\mathbb{C}) | A \cdot A^T = Id \} $ and ...
1
vote
0answers
52 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
1
vote
0answers
36 views

Reference request: exact sequences of Lie algebras

I have a reference request: where can I read more about the following? Consider the short exact sequence $0\rightarrow \mathfrak{n}^- \rightarrow \mathfrak{gl}_n\rightarrow \mathfrak{b}\rightarrow ...
1
vote
1answer
62 views

Reference for Weyl Character Formula

I am reading Lie algebra book by James E.Humphreys. This book giving enough discussion about Weyl Character Formula and its proof, still I would like to know what are the other books or lecture notes ...
0
votes
1answer
38 views

The group $\mathrm{SL}(n,\mathbb{C})$ .

I am looking for a detailed study of the group $\mathrm{SL}(n,\mathbb{C})$, its topology, subgroups, on any point of view (algebraic, complex or real Lie group theory, etc.). I am interested in any ...
0
votes
2answers
86 views

What are the good textbooks on Kac-Moody groups?

While there is a number of good books on Kac-Moody algebras ("Infinite dimensional Lie algebras" by Kac is already enough), it seems to me there is lack of textbooks on Kac-Moody groups. nLab says ...
1
vote
1answer
34 views

Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation

Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me: as a quotient of a semisimple real Lie group $G$ ...
0
votes
0answers
30 views

References request about exponentials in Lie algebras.

I saw two formulas about Lie algebras. Let $G$ be an algebraic group over $k$ and $\mathfrak{g}$ its Lie algebra. For any $x \in \mathfrak{g}$, $a \in k$ and $g \in G$, we have $$ g \exp(ax) g^{-1} = ...
2
votes
2answers
63 views

Characterization of differentiability via Lie derivatives

Yesterday I asked this question in MathOverflow but did not receive an answer yet. I want to try my chance here too, since I am in kind of a hurry. Answers will be much appreciated. I intend to ...
4
votes
0answers
41 views

Analytic/Smooth/Continuous maps between a manifold and itself

Let us suppose that $M_{\omega}$ is a connected real-analytic manifold of dimension $n$. Then there is an associated smooth structure, $\mathcal{C}^r$ structure ($r$ non-negative integer) on it. Let ...
3
votes
1answer
60 views

$p$-adic analytic group are closed subgroups of $GL_n(\mathbb{Z}_p)$ for some $n$

The article on pro-$p$-groups on Wikipedia tells us, that any $p$-adic analytic group can be found as a closed subgroup of $GL_n(\mathbb{Z}_p)$ for some $n \geq 0$. Do you have a reference for that ...
0
votes
0answers
18 views

Continuous groups of Transformations [Reference request]

I am considering reading the book : 'Continuous Groups of Transformations' by Luther Pfahler Eisenhart. It seems to have a very interesting table of contents. However this is quite old and I am ...
10
votes
1answer
111 views

Good book for studying $S_\infty$.

I'm looking for any books with some good information involving $S_\infty$ and other Polish groups. Specifically interested in $S_\infty$. This is an extremely amazing topological group, now having ...
3
votes
2answers
142 views

Reference request for studying Lie group & Lie algebra representations

I am learning representation theory of Lie groups & Lie algebras from the book by Brian Hall. Unfortunately, this does not discuss infinite dimensional representations. Which books should I study ...
1
vote
0answers
74 views

Sources for learning Lie groups and symplectic geometry for Quantum optics

I am asking this question on behalf of my junior who has recently joined in the graduate programme. As suggested by my boss, the student wants to work on quantum optics from a symplectic geometric ...
0
votes
0answers
68 views

Closed Lie subgroups of $SU(3)$

I'm looking for a reference describing the closed connected Lie subgroups of $SU(3)$. I know they are $SU(2)\times U(1)$, $SU(2)$, $SO(3)$ and several abelian subgroups based on this mathoverflow ...
3
votes
1answer
125 views

What are the one-parameter subgroups of GL?

Are the multiplicative one-parameter subgroups of the general linear group (i.e., morphisms $\lambda:\Bbbk^\times\to\mathrm{GL}_n\Bbbk$ of algebraic groups) completely classified? The obvious ...
1
vote
0answers
23 views

(Reference Request) Canonical forms for Real and Complex binary forms of low degree.

I am asking for a reference for Canonical forms for Real (and Complex) binary forms of low degree with respect to the natural action of the Real (and Complex) special linear group $SL_{n}(\mathbb{R})$ ...
8
votes
5answers
212 views

Getting started with Lie Groups

I am looking for some material (e.g. references, books, notes) to get started with Lie Groups and Lie Algebra. My motivation is that I (eventually) want to understand the theory underpinning papers ...
11
votes
1answer
153 views

Conditions for a topological group to be a Lie group.

In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157): Let $G$ be a locally compact ...
0
votes
0answers
180 views

References for basic level Differentiable Manifolds and Lie Groups

I an undergraduate math student with a decent background in abstract algebra. I am looking forward to studying Lie groups this summer...I want some you to suggest good references for the following ...
8
votes
1answer
324 views

Learning Roadmap for Borel - Weil - Bott Theorem

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have ...
13
votes
1answer
201 views

Why is the dimension of $SL(2,\mathbb{H})$ equal to $15$?

Let me ask a very basic question which is inspired by reading M. Atiyah's "Geometry and physics of knots". Could you explain me (or give a reference to) the definition of the special linear group ...
1
vote
0answers
25 views

How to write down the maximal subgroups of $GL(9, \mathbb{C})$

I am wondering about the maximal subgroups of the group $GL(n^2, \mathbb{C})$. My motivation for wondering about these groups is a project (in its most general form) I am working on where I am trying ...
2
votes
0answers
42 views

Usage and determination of “rank” and “dimension” of groups & representations

Physicist here. I seem to see conflicting statements about the rank of some groups I've come across lately. A paper I'm reading states that $SO(6)$ is rank 3 and therefore its Cartan subalgebra ...
2
votes
0answers
62 views

Equivariant homotopy equivalence of based loop group

Consider a compact, connected, simply connected Lie group $G$ and consider $S^1$ as an additive group. Let $\Omega G = \{ \gamma: S^1 \to G: \gamma(0) = e_G\}$ be the corresponding based loop group of ...
1
vote
0answers
58 views

Is the Hilbert-Smith conjecture still unsolved?

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group. Is this conjecture still unsolved? Is ...
7
votes
2answers
322 views

Which Lie groups have Lie algebras admitting an Ad-invariant inner product?

I am trying to answer the following question: Which Lie groups have a Lie algebra admitting an $\text{Ad}$-invariant inner product? First of all, all compact Lie groups satisfy this condition ...
9
votes
1answer
263 views

The center of a simply connected semisimple Lie group

I am learning about Lie groups, and I have the following basic question: Every Lie group $G$ has a (unique) universal covering group $ \bar G $ that is simply connected, and such that the covering ...
2
votes
1answer
316 views

Conjugate Representations

Are there any general results on when conjugate representations of a real Lie algebra are equivalent? I'm inclined to say that they are often not, but this is merely going on my case by case ...
5
votes
2answers
149 views

Classification of irreducible representations via Casimirs

Physicists almost always label irreducible representations via Casimirs (e.g., characterizing the irreducible representations of $SO(3)$ by spin). I've been looking far and wide to see the general ...
4
votes
2answers
404 views

$SU(2)$ Representation of $SO(3)$

I've often seen it written that $SU(2)$ is a "two-valued representation" of $SO(3)$ (in theoretical physics books mainly). I have a major conceptual issue with this however. I know there is a Lie ...
3
votes
1answer
73 views

Dimension of the GL-orbit of d-forms in one less variable

Let $V:=k[x_0,\ldots,x_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$. Let $G:=\mathrm{Gl}(n+1,k)$ act on $V$ induced by the canonical action on the linear forms: For ...
21
votes
10answers
3k views

What's a good place to learn Lie groups?

Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to ...
11
votes
2answers
737 views

References on Linear Algebraic Groups/Lie Theory

I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the ...
2
votes
0answers
136 views

Standard parabolic Lie subalgebras and conjugacy

Let $\mathfrak g$ be a given semisimple Lie algebra with corresponding adjoint Lie group $G$. A parabolic subalgebra is any subalgebra containing a Borel subalgebra. We can pick a Borel ...
4
votes
1answer
184 views

Parabolic subgroups of $\mathrm{Sl}_n$ are the ones that stabilize some flag

I am looking for a reference for the above statement that every parabolic subgroup of $\mathrm{Sl}_n(\Bbbk)$ stabilizes some flag in $\Bbbk^n$. I have gone through a large pile of books and can't seem ...
11
votes
0answers
165 views

Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$? $\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
3
votes
1answer
170 views

Why is $\mathbb{C}^{g}$ the universal cover of any connected compact complex Lie group of dimension g?

This question came up while I was studying the book "Complex Abelian Varieties" by Lange/Birkenhake. More precisely, the authors prove in Lemma 1.1 that every connected compact complex Lie group of ...
3
votes
3answers
187 views

the transformation which rotates a matrix by a half turn

Consider $$T_{2}: \left[ \begin{array}{cc} a & b \\ c & d \\ \end{array} \right] \rightarrow \left[ \begin{array}{cc} d & c \\ b & a \\ \end{array} \right] $$ $$T_{3}: \left[ ...
6
votes
1answer
155 views

Possibilities of an action of $S^1$ on a disk.

I'm dealing with actions of the circle over differentiable manifolds. In the book I'm reading, they use the fact that an action of $S^1$ over a disk has to be equivalent (there has to exist an ...
0
votes
2answers
121 views

What is the generalization of Lie group of transformation?

What is the generalization of Lie group of transformation? I found $a_1x+a_2$ and $(a_1x+a_2)/(a_3x+a_4)$ are also called Lie group of transformation!! It contradicts with what we learn about the ...
1
vote
1answer
47 views

reference in Montgomery/ Zippin

In a paper, the authors use the reference [M–Z, Theorem in 4.13] where [M-Z] denote the book D. Montgomery and L. Zippin, Topological Transformation Groups. Interscience Publishers, New York–London, ...
7
votes
0answers
304 views

Klein's Erlangen program taken seriously

Felix Klein suggested in his Erlangen program a way to classify geometries based on group theory. According to Wikipedia, we have the following definition: A Klein geometry is a pair (G, H) where ...
1
vote
2answers
257 views

Connections of Geometric Group Theory with other areas of mathematics.

I'm a master's student in the Turin University. At the end of my studies, I have to write a master thesis. My main interest is geometric group theory, but it is not a research area of the Turin's ...