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 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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7 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
19 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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11 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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32 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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27 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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21 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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32 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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20 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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12 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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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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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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50 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
45 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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19 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
55 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 ...
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1answer
42 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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15 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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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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50 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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21 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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62 views
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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
53 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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23 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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82 views

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
32 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
33 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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21 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 ...
2
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1answer
42 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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20 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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24 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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22 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
24 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 ...
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2answers
65 views

Why does exponentiating the derivative yield the shift operator?

If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
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Show that a homomorphism of Lie goups f s.t. f and def are injective is a Lie Subgroup.

My definition of a Lie subgroup for a Lie group $G$ is a submanifold $f : H \rightarrow G$ s.t. f is a homomorphism. So I only need that for each $x \in G$, $d_xf$ is injective and I'm done. So since ...
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1answer
23 views

If the Lie algebra is a direct sum, then the Lie group is a direct product?

I am reading the corollary 21.6 in the book "Morse Theory" by John Milnor, but I've encountered a statement for which I have no ideas. Let $G$ be a simply connected Lie group with a bi-invariant ...
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1answer
45 views

Representation theory and particle physics

Are there good books which explain clearly explain the connections between modern particle physics and representation theory of groups and lie algebras?
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16 views

Cartan subalgebra of product

i have a simple question what is the Cartan subalgebra of Lie algebra associated to the Lie group ?
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1answer
33 views

Advice on proving a tricky inequality

Im a little out of my depth here and am not well versed in combinatorics. Im not sure if this problem is too hard to solve or if there exists well known results to prove it. Here is part 1 which might ...
3
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1answer
151 views

Definition of Lie Groups

In the definition of Lie Group, we require that $$(x,y)\rightarrow x*y \text{ and } x\rightarrow x^{-1}$$ both be smooth. Are there any examples of groups that satisfy only one of these and not the ...
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30 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.
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28 views

Differential equations and groups

Consider the following linear ODE $$\frac{d\vec{x}}{dt} =A \vec{x}$$ where $A$ is an invertible matrix. This is the motivation of the question. I wondering whether $\{e^{At}\mid t\geq 0\}$ is a ...
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21 views

Universal Cover of $SL_2(\mathbb{R})$

I know that there is a way to define a multiplication map on a universal cover of a Lie group given in this post. However, I was wondering if there is a way to write this multiplication explicitly ...
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59 views

Is there a name for the group of complex matrices with unimodular determinant?

Does the group $$ G = \left\{ A \in \mathbb{C}^{n \times n} : |\det(A)| = 1 \right\} $$ have a name? It obviously contains the unitary group $U(n)$ and the special linear group $SL(n,\mathbb C)$. ...
3
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1answer
54 views

Building invariants of non-fundamental $SU(2)$

Suppose you have two objects, $ \phi _i $ and $ \psi _j $ that form representations of $ SU(2) $. With both fields in the fundamental representation, I believe there is one invariant, \begin{equation} ...
3
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1answer
45 views

Subtori of a complex torus

I am beginning to read Birkenhake-Lange, Complex Abelian Varieties, where they define a complex torus as being a quotient $X=\mathbb{C}^g/\Lambda$, where $\Lambda$ is a lattice in $\mathbb{C}^g$. ...
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44 views

Invariance under conjugation, equivalent in Lie Group and Lie Algebra?

Is the following true? $ e^X Y e^{-X} = Y \Leftrightarrow [X,Y]=0$ . From right to left you can show it with a corollary from the Baker–Campbell–Hausdorff formula. But in the other direction? I ...
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2answers
38 views

The discription of abelian Lie groups

There is a problem in my problem sheet to discribe all abelian connected Lie groups (moreover this is the first problem and it should be rahter easy). First it is difficult to understand how this ...
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28 views

Name of a Lie group

Does the Lie group $$G= \left\{ \left. \left( \begin{array}{cccc} x & y_{1} & \cdots & y_{n-1} \\ 0 & 1 & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & \cdots ...
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16 views

Combining infinitesimal generators of diferent dimensions

I am reading a paper about ways in which you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of $SU(3)\times{}SU(2)\times{}U(1)$. At a certain point, it starts considering ways of getting ...