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 (group-...

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38 views

Compute $ad_X$, $ad_Y$, and $ad_Z$ relative to a basis

For a lie algebra $\mathbb{g} $ we can define the adjoint representation as: $ ad: \mathbb{g} \rightarrow End(\mathbb{g}) $ as the map such that $ad_x(y)=[x, y] $ for all $\in \mathbb{g} $ I am ...
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567 views

Precise connection between complexification of $\mathfrak{su}(2)$, $\mathfrak{so}(1,3)$ and $\mathfrak{sl}(2, \mathbb{C})$

I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the ...
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2answers
256 views

Subgroups of $\Bbb{R}^n$ that are closed and discrete

I am trying to prove that every closed discrete subgroup of $\Bbb{R}^n$ under addition is a free abelian group of finite rank. I have tried to do this by induction on the dimension $n$. Base ...
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290 views

The generators of $SO(n)$ are antisymmetric, which means there are no diagonal generators and therefore rank zero for the Lie algebra?

Okay, this may be a silly question but I can't figure it out myself right now. By definition $O \in SO(n)$ fulfils $O^T O=1$ and $\det(O)=1$. For the generators of the group $ T_a \in so(n)$, this ...
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217 views

Defining an isomorphism that respects the Lie bracket: is my work correct?

I previously determined that if $\mathfrak{sl}$ denotes the Lie algebra of $SL_2(\mathbb C)$ and $\mathfrak o$ denotes the Lie algebra of $O(3,\mathbb C)$ then a basis for $\mathfrak{sl}$ is given by ...
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1answer
225 views

Help needed in understanding the basics of Cartan decomposition of a Lie algebra

I am trying to learn the basics of Cartan decomposition of Lie algebra, and have come across the following example. Consider $\mathfrak{gl_n}$ as the Lie algebra of endomorphisms of $\mathbb{C}^n$. ...
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574 views

Differential of the inversion of Lie group [duplicate]

Let $G$ be a Lie group and $\iota \colon G\to G$ denote the inversion. If $e$ is the identity of $G$, prove that: $$\text d \iota _e = -\text {id} _{T_e G}.$$ I understand that the differential at $e$ ...
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144 views

Is trace of regular representation in Lie group a delta function?

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
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1answer
102 views

Where does the ambiguity in choosing a basis for a Lie algebra come from?

This is a follow-up to this question. For matrix Lie algebras, we can define the Lie algebra $g$ of a group $G$ as the set $T_a \in g$ that yield an element of $G$ when put into the exponential map: ...
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1answer
36 views

Radical of a Lie algebra bracket itself

Let $\mathfrak{g}$ be a Lie Algebra over $k$, $\mathfrak{n}$ its radical. Why is $[\mathfrak{n},\mathfrak{g}]$ the smallest of its ideals $\mathfrak{a}$ such that $\mathfrak{g}/\mathfrak{a}$ is ...
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64 views

Let $\rho$ f.d. rep of a nilpotent Lie algebra such that $\rm{det} \rho(X) = 0$, $\forall X$. Then $\exists v \neq 0$: $\rho(X)v = 0, \forall X$.

Could you help me with this question? Let $\mathcal{g}$ be a nilpotent Lie algebra over $k$ and $\rho$ a representation of $\mathcal{g}$ in a finite-dimensional nonzero vector space $V$ over $k$. ...
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431 views

A covering map from a differentiable manifold

Let $p: C \to X$ is a covering map. Suppose that $C$ is a differentiable manifold. Is X - differentiable manifold? More precisely, I am interested in the case where $C$ is Submanifold of Lie algebra, ...
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1answer
80 views

Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)\times SU(2)$, or their Lie algebras

I have seen sources claim that $SO^+(1,3) \cong SU(2) \times SU(2)$, but have seen others claim that only their Lie algebras are isomorphic. Is it true that $SO^+(1,3) \cong SU(2) \times SU(2)$? If ...
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317 views

Why is $\mathfrak{sl}(n)$ the algebra of traceless matrices?

I'm studying Lie algebras as purely algebraic objects, without much of a background in the differential geometry surrounding Lie groups. The definition of $\mathfrak{sl}(n)$ has been given to me as ...
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1answer
124 views

Derivative of a group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ where $g: I \rightarrow G$ and $d: I \rightarrow M$ are ...
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Given a group $ G $, how many topological/Lie group structures does $ G $ have?

Given any abstract group $ G $, how much is known about which types of topological/Lie group structures it might have? Any abstract group $ G $ will have the structure of a discrete topological group ...
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1k views

Lie algebra of a quotient of Lie groups

Suppose I have a Lie group $G$ and a closed normal subgroup $H$, both connected. Then I can form the quotient $G/H$, which is again a Lie group. On the other hand, the derivative of the embedding $H\...
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1answer
209 views

Harmonic Analysis on the real special linear group

I would like to understand the representation theory and generalized Fourier transform of $SL(3, \mathbb{R})$ in as concrete a manner as possible. My ultimate goal is to develop an algorithm that can ...
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Recovering the two $SU(2)$ matrices from$ SO(4)$ matrix

Since there is a $2$-$1$ homomorphism from $SU(2)\times SU(2)$ to $SO(4)$ there should be a way to recover the two $SU(2)$ matrices given an $SO(4)$ matrix. I believe I could set this up as a ...
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770 views

Conditions for a smooth manifold to admit the structure of a Lie group

As we know, Lie group is a special smooth manifold. I want to find some geometric property, which is only satisfied by the Lie group. I only found one property: parallelizability. Can you show me ...
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241 views

Non-isomorphic Group Structures on a Topological Group

Which Topological Groups Have a Unique Group Structure (up to isomorphism)? I know that there are many non-isomorphic finite groups of same order, so there are many group structures possible for ...
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Description of SU(1, 1)

For a homework question, I am required to "describe the Lie group SU(1, 1)". This is a bit ambiguous, but I think what that means is I need to find a parametrisation of the elements of the group. I ...
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4k views

Intuitive explanation of Left invariant Vector Field

Intuitively what is meant by a left invariant vector field on a manifold?
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217 views

Is there a group between $SO(2,\mathbb{R})$ and $SL(2,\mathbb{R})$?

Is there a non-trivial subgroup $H \subset SL(2,\mathbb{R})$ such that $H \supset SO(2,\mathbb{R})$ ? My intuition is that, since $\dim SO(2)=1$ and $\dim SL(2)=3$, there should be some group between,...
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914 views

Calculating the Lie algebra of $SO(2,1)$

I am trying to calculate the Lie algebra of the group $SO(2,1)$, realized as $$SO(2,1)=\{X\in \operatorname{Mat}_3(\mathbb{R}) \,|\, X^t\eta X=\eta, \det(X)=1\},$$ where $$\eta = \left ( \begin{array}...
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238 views

Homotopy groups of some magnetic monopoles

This is a list of homotopy groups which I (as a physics researcher) encounter when studying magnetic monopole under certain configuration of gauge field profiles. \begin{gather} \pi_2(SU(2)/U(1)) \...
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1k views

Example of two-dimensional non-abelian Lie algebra?

can some one give me an example of two-dimensional non-abelian Lie algebra?
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604 views

Minimization on the Lie Group SO(3)

Refering to a question previously asked Jacobian matrix of the Rodrigues' formula (exponential map). It was suggested in one of the answer to only calculate simpler Jacobian $\left.\frac{\partial}{...
21
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1answer
526 views

Shrinking Group Actions

Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...
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412 views

Can we prove that there are countably many isomorphism classes of compact Lie groups without appealing to the classification of simple Lie algebras?

It is a nontrivial fact that there are only countably many isomorphism classes of compact Lie groups. One can prove this by a series of reductions: first to the connected case, then to the simply ...
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877 views

Fundamental group of $SO(3)$

How can I show that the universal cover of $SO(n)$, for $n\ge 3$, is a double cover? And how does that reflect the fact that the fundamental group of $SO(n)$ has two elements? What is the relation ...
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2answers
940 views

Homology and Euler characteristics of the classical Lie groups

I'm interested in methods of computing the homology groups and Euler characteristics of the classical Lie groups ($GL(n,\mathbb{R}), SL(n,\mathbb{R})$, etc.). (But I'd be interested in techniques ...
11
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2answers
537 views

Relation between $SU(4)$ and $SO(6)$

This is more of a particle physics question than maths. Since $SO(6)$ and $SU(4)$ are isomorphic, how are the fields (say for example scalar fields of ${\mathcal{N}}=4$ Super Yang Mills in $4d$) ...
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1answer
117 views

Flows and Lie brackets, $\beta$ not a priori smooth at $t = 0$

Let $X$ and $Y$ be smooth vector fields on $M$ generating flows $\phi_t$ and $\psi_t$ respectively. For $p \in M$ define$$\beta(t) := \psi_{-\sqrt{t}} \phi_{-\sqrt{t}} \psi_{\sqrt{t}} \phi_{\sqrt{t}}(...
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1answer
1k views

What are defining & fundamental representations?

In physics terminology, one hears of the fundamental & defining representations of lie algebras or groups - are these the same as irreducible representations?
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343 views

Visualizing Lie groups.

I like to visualize lie groups as flows on some manifold. For example: $SO(2)$ can be visualized as rotations of $S^1$ and it's lie algebra as constant vector fields on $S^1$. Or $SO(1,1)$ can be ...
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1answer
738 views

Expression for the Maurer-Cartan form of a matrix group

I understand the definition of the Maurer-Cartan form on a general Lie group $G$, defined as $\theta_g = (L_{g^{-1}})_*:T_gG \rightarrow T_eG=\mathfrak{g}$. What I don't understand is the expression ...
6
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1answer
159 views

Possibilities of an action of $S^1$ on a disk.

I'm dealing with actions of the circle over differentiable manifolds. In the book I'm reading, they use the fact that an action of $S^1$ over a disk has to be equivalent (there has to exist an ...
5
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1answer
616 views

Every principal $G$-bundle over a surface is trivial if $G$ is compact and simply connected: reference?

I'm looking for a reference for the following result: If $G$ is a compact and simply connected Lie group and $\Sigma$ is a compact orientable surface, then every principal $G$-bundle over $\Sigma$ ...
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230 views

equation involving the integral of the modular function of a topological group

Let $G$ be a locally compact topological group and $H$ a closed subgroup. Choose a left Haar measure $d\zeta$ for $H$, and let $d\mu$ be any measure for $G$. Also let $f$ and $g$ be continuous ...
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1answer
298 views

How does Maurer-Cartan form work

I have seen similar post asking for interpretation of the Maurer-Cartan form, but I am still struggling to understand it, so let me try to work a specific example and pose a specific question. Let $G$...
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169 views

What does $GL_n(R)$ look like?

Exactly as in the title - what does the general linear group "look like" (you are free to interpret this however you like) as submanifold of $R^{n^2}$? What should I imagine when I think of it? (I ...
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72 views

Calculating the differential of the inverse of matrix exp?

Let $A(t)$ and $B(t)$ be two matrix-valued smooth function satisfying the equation, $B(t) = e^{A(t)}$. I need to express $\frac{dA(t)}{dt}$ in terms of $B(t)$. I know that there is a formula of Wilcox,...
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84 views

Lie group step in proof

Let $X_e,Y_e \in T_eG$ be vectors and $G = GL(n).$ Then the right translation is given by $Y_g = Y_e g$ and $X_g = X_e g.$ Now, I have a proof showing that $[X_e,Y_e] \in T_eG$ is the element ...
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1answer
231 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
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1answer
467 views

What are spinors mathematically?

In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing. There are essentially two frameworks for viewing the notion of a ...
7
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278 views

Lie Group Decomposition as Semidirect Product of Connected and Discrete Groups

I've believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in this Math Overflow thread, it is mentioned ...
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963 views

Physical interpretation of the Lie Bracket

I've come accross this physical interpretation for $ [X,Y] $ which I don't understand : Follow $X$ for some time $\epsilon$; Follow $Y$ for $\epsilon$; Follow -X for $\epsilon$; Follow -Y for $\...
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1answer
605 views

Representations of a non-compact group are labeled by its maximal compact subgroup?

I don't have much of any awareness about the representation theory of non-compact Lie groups but I bumped into it for my work. Is there some idea that the representations of a non-compact group are ...
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211 views

One more question about mapping quaternionic matrices into real matrices

Real matrices that lie in the image of the inclusion homomorphism $\rho_n: M_n(\mathbb C) \to M_{2n}(\mathbb R)$ are called complex linear real matrices. It is easy to see that a real matrix is ...