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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Lie group representatiom - quasi-equivalent representations

Let $T$ and $U$ be unitary representations of a conected simply conected nilpotent Lie group, such that all irreducible subrepresentations of $T$ and $U$ are the same. If $T$ and $U$ are finite, then ...
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33 views

Root space of a Semi simple group an LVS?

A semi-simple Lie group has a Cartan Subalgebra ($H$) (CSA) -an LVS, Dual to this CSA LVS is root space($H^*$), which is set funtionals that map elements of CSA to real numbers and hence useful in ...
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53 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 ...
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15 views

Dual basis to $e_{i+1}-e_{i} \in \ker ((1,1,…1)^\vee\in(\Bbb E^{n+1})^\vee)$

Studying the root system $A_n$ given by the simple roots $v_i:=e_{i+1}-e_i \in \Bbb E^{n+1}/\Bbb R(1,1,...,1)$ for $i = 1,...,n$, I came across the following dual basis: $v_i^\vee:= ...
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45 views

How to prove this property of a projective transformation?

The copy below is from this book: Sophus Lie, Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen, bearbeitet und herausgegeben von Dr. Georg Wilhelm ...
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25 views

Global section $$s:T^*G\to Sp(T^*G)$$

Let $G$ be a Lie group and then how can we define the global section $$s:T^*G\to Sp(T^*G)$$ Where $Sp(T^*G)$ means symplectic frame bundle of $T^*G$.
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33 views

Is there a name for the function $g(z)=\frac{1-e^{-z}}{z}$?

This function comes in connection with exponentiation for matrix Lie groups, e.g. in computing the derivative of $\exp()$ away from the identity, or $\exp()$ of something in the affine group ...
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14 views

$U(n)/Z(U(n))$ is isomorphic to $SU(n)/Z(SU(n))$

A problem that I have been working is withere $U(n)/Z(U(n))$ is isomorphic to $SU(n)/Z(SU(n))$. I believe the best way to approach this problem is to show that they are both Isomorphic to the same ...
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1answer
52 views

Decompose complex vector by SU(4)

This question is about to decompose (or reduce dimension) complex vector by $SU\left( 4 \right)$. Given any $4\times1$ complex vector $B$. We can build $a_i$,and matrix$\lambda_i,i=1\ldots n $, $a_i ...
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1answer
25 views

$T_1 \times T_2$ is a maximal torus?

I have been working on teaching myself matrix groups and I have come across a problem about maximal tori. If I have a torus, $T_1 \subset G_1 $ and it is the maximal torus and If I have a torus, $T_2 ...
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1answer
37 views

prove that the maximal torus of $SO(3)$ is the maximal torus of $GL_3(\mathbb{R})$.

I want to prove that the maximal torus of $SO(3)$ is the maximal torus of $GL_3(\mathbb{R})$. I want to use the theorem that every maximal torus of G equals $gTg^{-1}$ for some $g \in G$. But I am not ...
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40 views

Can the equality $e^{-tY}Me^{tY} = e^{tX}M $ be shown by showing it only to 1st order? (Lie representations)

We have that A and B belong to different representations of the same Lie group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations. $$A = ...
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45 views

Ranks of matrix Lie groups and Lie algebra of SU(1,1), SO(2,1)

I was trying to find out by Googling, but had no luck. Am I right in thinking that for the Lie GROUPS: rank SL(n,R) = n, rank SO(n,R) = n (not sure about this one), rank SU(n,C) = n-1 and ...
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20 views

Unimodular groups

Let F be a non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq (where q is a power of p). Is GL(n,F) ...
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33 views

Three linearly independent vector fields

How can one find three linearly independent vector fields on $S^1\times S^2$? I know that $S^1\times S^2 \cong SO_3( \mathbb{R})$, i.e. the set of orthogonal $3 \times 3$ matrices with determinant ...
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35 views

$U(n)$ and $U(1)\times SU(n)$ are not isomorphic Lie groups

I am reading John Lee and on the chapter about group actions there is a problem that asks me to show that $U(n)$ and $U(1)\times SU(n)$ are not isomorphic Lie groups by showing that they don't have ...
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62 views

How can we extend the proof that Lie groups are orientable to parallelizable manifolds?

Let $G$ be a Lie group with identity element $e$ and dimension $n$. I know we can take a non-vanishing $n$-form (call it $\omega$) on the vector space $T_eG$. For a Lie group, as we have an ...
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2answers
44 views

Lie bracket of vector fields on $R^2$

Compute the Lie bracket$$\Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big]$$ on $R^2$ Can you help me please?
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14 views

About representations and transformations under an $SU(n)$ Lie Group

I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the ...
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1answer
31 views

Obtaining representations of $G$ from $\mathrm{Lie}(G)$.

Suppose $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$, and $\tilde{G}$ is the unique connected, simply connected Lie group whose Lie algebra is $\mathfrak{g}$. Let $C$ be any discrete ...
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26 views

Why is the Compact Symplectic Group Simply Connected

Let $Sp(n)=U(n,\mathbb{H})=\{A \in M_{n}(\mathbb{H}) : A\cdot A^{*}=I\}$ be the compact symplectic group, a subset of $Sp(2n,\mathbb{C})$. I want to show that $Sp(n)$ is simply connected, in ...
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58 views

Question about vector fields and Lie group

Notation: $\chi(G)$ is the set of smooth vector fields on Lie group $G$, which in fact forms a vector space. Given a Lie group $G$, show that there exists a smooth vector field $X\in \chi(G)$, ...
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48 views

Lie algabra of R^n

Until now the only example of lie groups I have seen are subgroups of $GL_n$. Today I had the idea, that also $G=(\mathbb R^n,+)$ must be a lie group ($(\mathbb R^n,+)$ is a group with the ...
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1answer
24 views

SO(n) is parallelizable

Prove that $SO(n)$ is parallelizable. How would I go about showing this? My supervisor could not help me with this problem, and I am stumped.
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1answer
36 views

In which course one learns Lie Group&Algebra and to which category of mathematics this subject belongs?

I'm a junior and i have never leanred this subject. I think "Lie Group&Algebra" is really deep and massive theory since the wikipedia page for it is quite long. Nevertheless, i'm not sure ...
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24 views

Rigidity for Lie Groups

This may be a very dumb question but I was wondering if the following train of logic is correct: We know a connected Lie group $G$ is isomorphic to the quotient $G\cong \tilde{G}/\Gamma$ where ...
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16 views

How could we decompose anticommutator of representation matrices for a Lie algebra?

For commutator, we know that $[T^a,T^b]=if^{abc}T^c$, where $f^{abc}$ is the structure constant. But is there a similar formula for $\{T^a, T^b\}$? Thank you.
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47 views

Is $(\mathbb{R},+)$ a smooth manifold?

I feel like $(\mathbb{R},+)$ is, but I'm not really sure. How would I know whether or not it is?
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29 views

Nontrivial relations in rotation groups

Consider the subgroup $H$ of $SO(3)$ generated by rotations of order $5$ (i.e., rotations by $\frac{2\pi}5$) about the $x$ and $y$ axes. This group certainly isn't finite or discrete (as it's not ...
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1answer
61 views

Centralizer of $SO(n)$

Given the set $M(n,\mathbb C)$ of all complex $n\times n$ matrices, what's the centralizer of $SO(n)$ in $M(n,\mathbb C)$? For $n=2$, the centralizer must be the matrices $A$ such that $RA=AR$ where ...
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1answer
57 views

The Lie derivative by an infinitesimal action is invertible at an isolated zero point

Let a compact Lie group $G$ act on a manifold $M$. Fix $X \in \mathfrak g$ and we write $X_M$ for the infinitesimal action of $X$. Assume that $p \in M$ is a zero point of $X_M$. Define a linear map ...
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1answer
26 views

Show the Hermitian matrices, with trace(g*g1,1)=0 form a vector space.

This is a question from an example sheet that I think may have a mistake in it. Show that the set of Hermitian matrices $A \in H_2 (\mathbb{C})$ with Trace$(A\cdot A_{(1,1)})=0$ is a real three ...
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2answers
71 views

Action of $SO_n$ on $\mathbb{S}^{n-1}$ induces fibre bundle.

Real compact Lie group $SO_n$ acts smoothly and transitively on $\mathbb{S}^{n-1} \subseteq \mathbb{R}^n$ with obvious action. Isotropy subgroup of each point in $\mathbb{S}^{n-1}$ is isomoprhic to ...
2
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1answer
44 views

Homotopy Type of Diffeomorphism Group of Lie Group

Let $G$ be a finite dimensional connected Lie group and $Diffeo(G)$ be the diffeomorphism group of the underlying manifold. Is it true that $Diffeo(G)$ has the homotopy type of a finite dimensional ...
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2answers
16 views

orthogonal transformation, $S^{n-1}$

Show we can find an orthogonal transformation of determinant $1$ sending any point of $S^{n-1}$ into any other. I searched online and could not find the solution. I am currently learning about Lie ...
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26 views

Learning representation theory of real reductive lie groups

I am interested in any sources that can be helpful for learning the representation theory of real reductive groups. I am currently reading Wallach book, but I feel that I don't understand the subject ...
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1answer
87 views

What is the kernel of a Maurer-Cartan form?

The Maurer-Cartan form on the Lie group $Gl(n,\mathbb{R})$ is a one-form taking values in $\mathfrak{gl}(n,\mathbb{R})$ as defined in the link. It has a rather concrete "extrinsic definition" as ...
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34 views

Lie group - exponential Diffeomorphism

Let $G$ be a nilpotent, connected simply connected Lie group and $\mathfrak{g}$ its Lie algebra. It is known that the exponential map $\exp$ is a diffeomorphism. Now let $\mathfrak{g}_0$ be a Lie ...
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1answer
78 views

Isogenies and dimensions

Let $f: \mathbb{C}^g/L\to\mathbb{C}^{g'}/L'$ be an isogeny of complex tori, i.e. a surjective Lie group morphism with finite kernel. Is it obvious that $g\ge g'$ ? It is easy to show that $f$ is ...
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2answers
55 views

If a group is isomorphic to a Lie group will that group also be a Lie group?

So I was working through some exercises on Lie groups and I was wondering if the group isomorphisms carry any of the differentiable structure with them. Explicitly, if two groups, $G$ and $H$ are ...
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28 views

Automorphism Groups of Lie Groups

Take $X$ to be a Lie group and $Aut(X)$ to be its automorphism group (group isomorphisms which are also homeomorhisms). In general, are there some Lie groups in which this can be computed? For ...
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The natural representation of $SO(n)$ is irreducible for $n\ge 3$

The natural representation $(\pi,\mathbb C^n)$ of $SO(n)$ is the one for which $$\pi (g)z = g^{-1}z$$ for $g\in SO(n)$ and $z \in \mathbb C^n$ (the product $g^{-1}z$ is just the usual matrix ...
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3answers
38 views

A question on the codimension of $O(n)$ and $SO(n)$ reative to $GL(n,R)$

This seems like a very silly way to control the website, because as a new user it will not let me make comments. So apologies in advance to reference a previous question, but given the warning not to ...
2
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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. ...
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27 views

Lie Group - derivatives

This is really a simple question. Let $A$ be an associative, nilpotent real algebra, and set $[a,b]=ab-ba$, define the exponential map as usual, that is $exp(a)=1+a+\frac{a^2}{2}+...$. Let ...
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1answer
47 views

Lie Automorphisms

Take $X$ to be a Lie group. Define a Lie automorphism of $X$ to be a group isomorphism from $X$ to itself which is also a homeomorphism. Define $Aut(X)$ to be the group of Lie automorphisms of $X$ ...
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24 views

What is the classification theorem of simple Lie groups?

I've seen this thrown around a bit, but I can't find what the theorem actually states? Can anyone help?
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27 views

Volume form for $SE(n)$ and/or $E(n)$. [duplicate]

I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the riemannian ...
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1answer
30 views

General Linear Group is(not) compactly generated

We know that any connected Lie Group is compactly generated. I have a feeling that $\mathbb{GL}_n\mathbb{R}$ is not compactly generated. Is it true? If it is how can I prove this? If not, what ...
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1answer
26 views

Showing a simple Lie group is connected and compact.

I'm working on a presentation on simple Lie groups and would like to show by example that the simple Lie groups are connected, but I'm not really sure how to do this. I'd also like to show that one of ...