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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10 views

How to compute $df(e)$ explicitly?

I am reading the book. On page 244, the formula (9.2.3.4). I would like to compute the bracket on g^* induced from the Poisson bracket on C[G] explicitly in the example of $G=SL_2$. The formula is: ...
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30 views

Is $U(1)$ a normal subgroup of $U(2)$ and does the question even make sense?

I have been wondering whether $U(1)$, defined as the group of complex phases (edit for clarity: I mean complex numbers of unit absolute value, such as $e^{i\alpha}$ with $\alpha \in \mathbb{R}$) with ...
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2answers
23 views

Ideals of $\mathfrak{gl}_n$

How does one determine the ideals of $\mathfrak{gl}_n(C)$? My guess is that the only ones are $(0) $ and $\mathfrak{sl}_n(C)$. I think approaching the problem by the fact that each $\mathfrak{g}^{ ...
2
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0answers
61 views

Proving a compact Lie group admits a biinvariant metric [duplicate]

At the end of a lesson in Differential Geometry, my teacher said: Fatto, che non dimostriamo, non è difficile ma il tempo scarseggia, se $G$ è compatto possiamo sempre trovare una metrica ...
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15 views

Levi-Civita connection with biinvariant metric

I'm struggling with the proof of the following, well-known result for the Levi-Civita connection of a Lie Group with biinvariant metric, i.e. satisfying \begin{equation} g(D_bL_a X_b, D_b L_a Y_b)= ...
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2answers
19 views

Are there multiple non-isomorphic principal $G$-bundles on Euclidean space? [duplicate]

I'm pretty sure the answer is out there, see this MathOverflow question, but that is unfortunately way over my head :). I'm interested in the case that $G$ is a Lie group (e.g. $U(1)$), but I don't ...
2
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0answers
25 views

Writing $\mathrm{SO}(2)$ as the zero-set of a function

Here I'm assuming $M_{2 \times 2}(\mathbb{R}) \cong \mathbb{R}^{4}$. The definition of $\mathrm{SO}(2)$ is: $\mathrm{SO}(2)=\{ \ A \in M_{2 \times 2}(\mathbb{R}) \ | \ \det(A)=1 \mathrm{\ and\ ...
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1answer
34 views

Center of $\mathfrak{gl}(n,\Bbb F)$ using the adjoint representation

I'm quite new to Lie Algebras, and so there's a lot of easy stuff that I'm probably missing. Anyway following Kac notes I'm asked to compute the center of $\mathfrak{gl}(n,\Bbb K)$, and I've done it ...
2
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0answers
35 views

Coset Space as a Representation of a Lie Algebra

I'm reading through some notes (about the use of Lie groups/algebras in physics) obtained from a friend from a class that took a while back, and I can't quite figure out where one thing regarding some ...
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1answer
52 views

Is $SL_n(\mathbb{R})$ actually simple?

It's probably not hard to prove that $\frak{sl}_n\mathbb{R}$ is simple, so that $SL_n\mathbb{R}$ has no nontrivial connected normal subgroups. But do there exist discrete normal subgroups of ...
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27 views

Are there proper Ad invariant sets on simple lie algebras?

Let $\mathcal C\subset \mathfrak g$ be a subset in a Lie algebra $\mathfrak g$ satisfying the following two conditions: $\mathrm{Ad}(G)\mathcal C=\mathcal C$ If $X,Y\in \mathcal C$, then ...
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1answer
33 views

$\mathrm{ad}$-invariant splitting of Lie algebras.

I am reading a text on which the following situation appears: Let $G$ be a a Lie group and $H$ a compact Lie subgroup. Let $\mathfrak{g}$ and $\mathfrak{h}$ be their Lie algebras respectively. The ...
4
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1answer
47 views

For which Lie groups $G$ can one write $g$ as the exponential $\exp X$ of some $X \in {\frak g}$ for every element $g \in G$?

I am reading a book on matrix Lie algebras (Brian Hall's). Corollary 2.30. says that if $G$ is a connected matrix Lie group, then every element $A$ of $G$ can be written in the form ...
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1answer
33 views

Proof that the special linear group $\mathrm{SL}(n,\mathbb{R})$ is a smooth manifold

This is my definition of a smooth manifold I am supposed to work with: Let $\mathcal{M} \subseteq \mathbb{R}^{n}$. The set $\mathcal{M}$ is a $k$-dimensional smooth submanifold of $\mathbb{R}^{n}$ ...
3
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1answer
16 views

Any bi-invariant distance on a group is inverse-invariant?

$\newcommand{\inv}{\text{inv}}$ Let $G$ be a Lie group. Assume $d$ is a metric on $G$ (in the sense of metric spaces) which is bi-invariant. Is it true that the inverse automorphism must be an ...
3
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1answer
72 views

Lie groups pre-requisites and reference

What are the minimum pre-requisites in analysis (differential geometry) required to study Lie-groups? And for that material, what are some good references? I have done basic courses in Metric spaces, ...
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6 views

Alternative construction of the twisted group $^2 E_6 (q)$.

I am looking for the alternative construction of the twisted finite simple group $^2 E_6 (q)$ possibly avoiding the Lie theory. Especially I am interested in calculating its order. Maybe some of you ...
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0answers
44 views

Some doubts on the relationship between Lie algebras and Lie groups

Let $(\mathbb G,*)$ be a Carnot group. Thus, by definition, $\mathbb G$ is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ admits a stratification, that is $$\mathfrak ...
3
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1answer
853 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 ...
0
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0answers
27 views

How to prove that a matrix lie group is (or is not) simple

I am having some difficulties in proving that some matrix lie groups are or are not simple. I am wondering what I should ask myself to solve this problem quickly. Should I look for normal subgroups? ...
3
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0answers
22 views

Pic of a variety of type G/P

Let $G$ be an simple algebraic group an let $P$ be a parabolic subgroup of $G$. Let $X$ be the projective, homogeneous variety $G/P$. Is it true that the following holds: Pic($X$) has rank $1$ iff ...
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0answers
6 views

To prove that a generator-candidate is sufficient to find all elements in $SO(3)$

I am attempting to prove that some sequential series of rotation axes $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n\in\mathbb{R}^3$ is enough to generate all possible rotations when making a full ...
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0answers
12 views

Understanding if the definition of constant normal set depends on the choice of the scalar product or not

Suppose we have a Lie group on $\mathbb R^n$, let's say $(\mathbb R^n,*)$. Suppose also that its Lie algebra $\mathfrak g$ is stratified: I mean that there exists a decomposition of $\mathfrak g$ as ...
4
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42 views

Open problems in Lie theory

I been studying lie theory for some time. Beside classification related problems what are some examples of open problems in the lie world? Especifically in the topological/differentiable structure of ...
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1answer
30 views

Examples of non linear Lie-Groups

When looking for a non linear Lie Group I always find the example of the Heisenberg Group $H$ modulo a normal Group $N$. Where the matrix of the two groups are of this form $$ H = \begin{bmatrix} 1 ...
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1answer
14 views

path connected componenet and connected components in a Lie group coincide

does anyone have an explanation\proof as to why path connected components and connected components of Lie groups coincide?
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27 views

Proof that $\exp(a) \cdot r \cdot \exp(-a) =\ exp(ad_a) \cdot r$ [closed]

Let a nilpotent element of associative algebra over field with zero characteristic and $ad_a: b \to [a, b] = ab -ba,$ $b \in R $. Proof that $\exp(a) r \exp(-a) = \exp(ad_a)r (r \in R) $
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1answer
113 views

Group of affine transformation in plane is unimodular

I am trying to do an exercise in the book "Analysis on Lie group" as follows: Let $G$ be the group of all affine transformations in the plane, i.e. $G$ contains all the mapping of form $(x,y)\mapsto ...
2
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1answer
33 views

Embedding of classical Lie groups

This is somehow very natural question so I'm sure that the answer should be well known: Whitney theorem states that each (say paracompact) $n$-dimensional manifold could be embedded in ...
5
votes
1answer
106 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
2
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27 views

Examples of non unimodular groups

I'm looking for criteria to know if a (lie) group is unimodular, the thing is that I only know one non unimodualr group that arrises naturaly, the affine group ax+b, and their direct generalizations. ...
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17 views

Action on Flag manifold

When $G$ is of type A,D,E and $B_4$ then the group of Dynkin diagram automorphisms is non-trivial. If $B$ is a Borel subgroup of $G$, then is there a nice action of the Dynkin diagram automorphism ...
4
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1answer
45 views

Is there a non-abelian Lie group which is homeomorphic to an $n$-dimensional torus $\mathbb{T}^n$?

I've learned that a compact connected abelian Lie group must be a torus. Of course, conversely, a torus as a group is abelian. I wonder if 'homeomorphic to a torus' is enough to imply abelian. ...
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1answer
27 views

Is the Lie algebra morphism induced by surjective Lie group morphism also surjective?

Let $G$ be a matrix lie group and $\Pi : G\to \Pi(G)$ a surjective Lie group morphism. Let $\mathfrak g$ and $\mathfrak h$ be the respective Lie algebras of $G$ and $\Pi(G)$. Then there is a unique ...
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1answer
40 views

Dimension of a finite irreducible algebraic group

Let $G$ be an irreducible algebraic group over the field $K$ of characterstic 0. Let $A=K[x_1,...,x_n]/I(G)$ be the coordinate ring and $K(X)=Q(R)$ be the quotient field of $A$. (Since $G$ is ...
3
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1answer
28 views

Another way of describing a maximal torus

Consider the Lie group $SU(2)$. A maximal torus for $SU(2)$ is $$T=\left\{\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{-i\theta}\end{pmatrix}:\theta\in{\Bbb R}\right\},$$ and its Lie algebra is ...
3
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1answer
49 views

Could characters in harmonic analysis be generalized into $S^2$?

Consider a locally compact abelian (LCA) group $G$. The start of commutative harmonic analysis is the fact that the collection of characters $\chi : G \to S^1$ (thought of as $S^1 = \mathbb{T} ...
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0answers
39 views

Matrix representation of a 6-dimensional Lie algebra

The question is about the matrix representation of the following 6-dimensional Lie algebra, with 6 generators $t_1,t_2,t_3,t_4,t_5,t_6$. This Lie algebra is nilpotent, non-abelian, non-reductive and ...
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0answers
12 views

is the stabilizer of the connected component of a liegroup contained in the connected component of the stabilizer?

Let $G$ be a non-connected Liegroup acting on a manifold $M$. For $x \in M$ we denote $G_x$ the stabilizer of the $G$-action on $x$. For a arbitrary Liegroup $K$ we denote by $K^\circ$ the connected ...
3
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1answer
73 views

When is a lattice dense in a torus?

Let $\pi: \mathbb{R}^n\rightarrow \mathbb{R}^n/\mathbb{Z}^n$. What (necessary and sufficient) criteria on $A\in GL_n(\mathbb{R})$ guarantee $\pi(A\mathbb{Z}^n)$ is dense?
5
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0answers
109 views

A Quotient of the Euclidean Group

$\newcommand{\euc}{\mathscr I}\newcommand{\R}{\mathbf R}$ Let $\euc(n)$ denote the the Euclidean group $\R^n\rtimes O_n(\R)$. Recall that $\euc(n)$ acts on $\R^n$ as $(\mathbf x, T)\cdot \mathbf ...
5
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3answers
187 views

Can a Compact Lie Group have a Non-Compact Lie Subgroup?

Hopefully the title says it all in terms of the question I'm asking. I feel that the answer is no, but I'm not sure why; I don't really know many deep theorems about the structure of lie groups.
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18 views

How is vector, dual vector, etc defined in Matrix Lie Group Manifold?

How is vector, dual vector, etc defined in Matrix Lie Group Manifold? Are the coefficients matrices and the (dual)basis matrices as well?For example, the Maurer–Cartan form can be written as ...
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0answers
14 views

Why are the vector fields of an homogeneous Lie group on $\mathbb R^N$ pyramid shaped?

I have a very precise question. I'm stuck on a stupid thing, at page 35 of Stratified lie groups and potential theory for their sub-Laplacians by Bonfiglioli,Lanconelli e Uguzzoni. In Remark 1.3.7 ...
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0answers
19 views

Lie element in a non-commutative algebra?

While I am reading a paper, there is a weird, at least for myself, notion I have never seen: let $R:=\mathbb{Q}_\ell\{\{X,Y\}\}$ be a $\mathbb{Q}_\ell$-algebra of formal power series in non-commuting ...
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1answer
22 views

Left multiplication isometry?

If $G$ is a semi-simple Lie group and $g\in G$, then $G$ has a bi-invariant metric which is a Riemmanian metric. My question is: with respect to this metric does the left-translation map ...
1
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1answer
109 views

Show exponential function maps any line through the origin onto a circle of radius 1 in $\mathbb{S}^3$.

Show that the exponential function maps any line through the origin in $\mathbb{R}i +\mathbb{R}j + \mathbb{R}k$ onto a circle of radius $1$ in $\mathbb{S}^3$. I know that for any element v ...
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0answers
16 views

Is this a fundamental domain of 2-torus under the action of Z2?

Let $U$ be a vector with relatively prime integer coordinates in $\mathbb{R}^2$. And let $V$ be another vector that is orthogonal to $U$ and the rectangle spanned by $U$ and $V$ are $1$. Is this ...
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2answers
50 views

Defining Unitary Matrices

I have got a question and I would appreciate if one could help. I start with an example to explain what I am looking for. Assume a scaled unitary matrix like $U_2 = \begin{bmatrix} 1 & 1 \\ 1 ...
1
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1answer
34 views

Compactification of Lie Group

Is there a way to embed a Lie Group $G$ into a compact lie Group $H$, such that the inclusion is a Lie group homomorphism?