# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $H\... 1answer 209 views ### Harmonic Analysis on the real special linear group I would like to understand the representation theory and generalized Fourier transform of$SL(3, \mathbb{R})$in as concrete a manner as possible. My ultimate goal is to develop an algorithm that can ... 3answers 3k views ### Recovering the two$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 ... 3answers 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 ... 2answers 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 ... 1answer 4k views ### 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 ... 1answer 4k views ### Intuitive explanation of Left invariant Vector Field Intuitively what is meant by a left invariant vector field on a manifold? 2answers 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,... 2answers 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}... 2answers 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)) \... 2answers 1k views ### Example of two-dimensional non-abelian Lie algebra? can some one give me an example of two-dimensional non-abelian Lie algebra? 3answers 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}{... 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 ... 1answer 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 ... 2answers 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 ... 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 ... 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) ... 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}}(... 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? 1answer 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 ... 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 ... 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 ... 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$\Sigma$... 2answers 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 ... 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$G$... 2answers 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 ... 2answers 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,... 2answers 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 ... 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$A$. ... 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 ... 3answers 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 ... 3answers 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$; Follow -Y for$\...
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 ...