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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40 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 ...
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31 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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53 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
68 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 $\mathbb{R}^{n}$ ...
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23 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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47 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
37 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 ...
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1answer
62 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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31 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? ...
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32 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
8 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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0answers
53 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
19 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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75 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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1answer
58 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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23 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 ...
3
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1answer
34 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 ...
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41 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 ...
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0answers
34 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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0answers
49 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
15 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 ...
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3answers
194 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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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
25 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 ...
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0answers
18 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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1answer
36 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?
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2answers
55 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 ...
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0answers
26 views

Maximal torus of a compact algebraic group

An algebraic group $G$ is a group object in the category of algebraic varieties, i.e. it is an algebraic variety with Zariski topology and group structures. Example for linear algebraic groups are ...
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1answer
41 views

Higher self-extension $\text{Ext}^i_{\mathcal{O}}(L(\lambda), L(\lambda))$ between two irreducible modules in BGG category $\mathcal{O}$

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $\mathcal{O}$ be the BGG category for $\mathfrak{g}$. It is well-known that the set of irreducible ...
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0answers
21 views

Differential equations and Vector spaces

I was reading Cohn's book on Lie Groups.In introduction part he has given the motivation behind Lie Groups.It is like this If solution of the differential equation $\frac{dx_{i}}{dt}=u(t)$ is ...
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2answers
47 views

Riemannian metrics on homogeneous spaces

Let G be a Lie group and H be a compact subgroup. The (left) coset space G/H is, up to an isomorphism, equivalent to the smooth homogeneous manifold M. My question is, is it possible to impose an ...
3
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1answer
51 views

Constructing an explicit non-contractible path in $\text{GL}_n(\mathbb{R})$

As can be seen here, the fundamental group of $\text{GL}_n(\mathbb{R})$ is $\mathbb{Z}/2\mathbb{Z}$ (for $n \ge 3$). (For $n=2$ it is $\mathbb{Z}$). Is there a way to find an explicit representing ...
2
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1answer
63 views

Regarding the representation theory of $SL_2(\mathbf{R})$.

Dear friends of mathematics, I have the following question for you. (a) According to Wikipedia there is a unique irreducible (real??) $2$-dimensional representation of $SL_2(\mathbf{R})$, which must ...
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1answer
24 views

How to show the space of inverse-invariant metrics on a Lie group is infinite dimensional?

Let $G$ be a Lie group. I am trying to convinve myself there are 'many' Riemannian metrics on $G$ for which the inverse automorphism is an isometry. Denote the iverse by $i$. For any metric $g$ on ...
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0answers
16 views

Liegroup acting transitively => connected component acts locally transitive?

Suppose $G$ is a Liegroup acting transitively on a manifold $M$. Does that already imply, that the connected component $G^\circ$ of $G$ acts transitively on the connected components of $M$? I have a ...
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1answer
16 views

algebra operations

If $[x]=x^{i}\sigma^{i}$, Find an alternative form for the product $$ x^{i}n^{j}\sigma^{k}\epsilon^{ijk}=x^{i}(\vec{n} \times \vec{\sigma})^{i} $$ that has to be more compact than $$ ...
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2answers
36 views

SU(2) is not isomorphic to $T^3$ [closed]

How can we prove that $SU(2)$ is not isomorphic to $\mathbb {S^1×S^1×S^1}$ by using the definition of $SU(2)$?
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0answers
16 views

G-invariant polynomials for a concrete G

The collection of matrices$$ \left( \begin{array}{lll} a^2 & 2ab &b^2 \\ ac & bc+ad & bd \\ c^2 & 2cd & d^2 \end{array} \right)$$ indexed by $a,b,c,d \in \mathbb{R}$ is a ...
5
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1answer
69 views

Lie groups where $x \mapsto x^2$ is a diffeomorphism?

In every Lie group $G$ the function $x \mapsto x^2$ is a local diffeomorphism in a neighbourhood of the identiy. (This is because its differential is: $v \mapsto 2v$ when considered as a map from ...
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0answers
19 views

Criterion for semi-simplicity of Lie algebra generated by vector fields

Suppose I have a finite collection of smooth vector fields $V:=\{V_1,...,V_k\}$ on a smooth manifold $M$. Moreover suppose that the Lie algebra $g$, generated by $V$ (where the Lie bracket is defined ...
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0answers
59 views

Lie Algebra SU(2)

Given a two dimensional Hilbert-space, $\mathcal{H}$, and a vector $\eta \in \mathcal{H}$, of this space, if $\eta$ transforms in SU(2) like this, $$\eta \rightarrow e^{(-i\alpha ...
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1answer
29 views

Cartan Lie Algebra of the Unitary Group $U(N)$?

I am having trouble understanding the Lie Algebra terminology. What is the Cartain Lie algebra of the unitary group $U(n)$? It must be in many textbooks, but they explain it very generally in terms ...
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2answers
55 views

Is Cartan subalgebra of Complex semisimple Lie algebra the maximal Abelian subalgebra? I found two places give the different answers.

In wiki https://en.wikipedia.org/wiki/Cartan_subalgebra Example 4, it says that Cartan subalgebra of complex semisimple Lie algebra is not maximal Abelian subalgebra. However in Brian C. Hall's ...
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1answer
18 views

Cohomology of complex Lie groups via compact form

Let $G$ be a compact Lie group. Let $G_{\mathbb{C}}$ be a complex Lie group such that there is inclusion $i: G \rightarrow G_{\mathbb{C}}$ of Lie groups. Moreover I require that differential of $i$ ...
3
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0answers
25 views

Minimal word length of factorization of invertible matrices into elementary matrices

Let $K$ be a field. As is well known, one can decompose every matrix $A \in GL(n,K)$ into a product of elementary matrices. By an elementary matrix, I mean a matrix which belongs to one of the ...
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1answer
18 views

dimension of space of origin-symmetric ellipsoids

I wonder how I can compute the dimension of the space of all origin-symmetric ellipsoids? What is the best approach?
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1answer
62 views

What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma}$?

In quantum mechanics we learn about the Pauli spin matrices: $$ \sigma_1 = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0\end{array} \right) \hspace{0.25in} \sigma_2 = \left( \begin{array}{cc} i ...
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15 views

Calculation of commutator of Lie algebra for affine linear maps

This problem was answered before, but I'm stack with a technical point. Let $G$ be the Lie group of linear polynomials under composition (that is, affine transformations), $$\{x \mapsto ax+b, a\neq 0, ...