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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22 views

Are there some new “function or even topic” in lie theory with special functions? [on hold]

Every one: I research in lie theory with special functions. But I saw a lot of research for most of the special functions and polynomials. I wish you could recommend a specific kind of these special ...
1
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1answer
21 views

Spheres as Symplectic Homogeneous Spaces

Does there exist a description of the odd dimensional spheres as homogeneous spaces of the symplectic group. For $S^7$ it seems to me that we should have $S^7 \simeq Sp(3)/Sp(2)$, but I can't make a ...
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0answers
26 views

Simply connected linear algebraic group

Following is what I understand regarding the simply connected linear algebraic groups afer reading some definition in Hochschild's 'Basic Theory of Algebraic Groups' : (I don't know about fundamental ...
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2answers
47 views

If the Lie algebra is ${\frak g}={\frak a}\oplus{\frak b}$ then the Lie group is $G=AB$?

Let $G$ be a connected Lie group and suppose that its Lie algebra ${\frak g}$ splits into a direct sum of ideals $${\frak g} = {\frak a}\oplus{\frak b}.$$ Let $A$ be the connected Lie subgroup of $G$ ...
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21 views

$so(4)$ is isomorphic to $so(3)+so(3)$.

Since every $4×4$ skew-symmetric matrix can be written uniquely as a decomposition $$\begin{bmatrix} 0&-a&-b&-c ...
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14 views

Derived Algebra is nilpotent implies the lie algebra is solvable [duplicate]

How does one show that Derived Algebra is nilpotent implies the lie algebra is solvable. My attempt: Let $L$ be such a Lie-algebra then $[L,L]$ is nilpotent so it is solvable. So $[L,L]^{(n)}=0$ for ...
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0answers
29 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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34 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
26 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}^{ ...
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20 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)= ...
2
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0answers
68 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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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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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 ...
1
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1answer
36 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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0answers
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 ...
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
34 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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11 views
+100

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
18 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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46 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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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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23 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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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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0answers
45 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
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?
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, ...
1
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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 ...
1
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0answers
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 ...
3
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1answer
30 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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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 ...
2
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0answers
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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0answers
40 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 ...
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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 ...
0
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1answer
23 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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0answers
17 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 ...
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?
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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 ...
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0answers
17 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 ...
2
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1answer
38 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 ...
2
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0answers
20 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 ...
1
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2answers
37 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
votes
1answer
44 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
62 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
22 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 ...