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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Intuition Lie Bundle

I am thinking about the following discretization problem: I want to rotate a given discrete 2D array over arbitrary angles around the origin, thus I want to be able to represent all rotated versions ...
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1answer
9 views

Is it always true that $N_{(G,G)}(T_1) \subseteq N_G(T)$?

Let $G$ be a connected, reductive linear algebraic group whose semisimple rank is $1$. Then $H := (G,G)$ is a connected semisimple group of rank one. Let $T_1$ be a maximal torus of $H$, and let $T$ ...
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28 views

Commuting derivatives in a Lie group

Let $G$ be a Lie group and $f = f(t, s) : \mathbb{R}^2 \to G$ smooth. Consider $\theta \in \Omega^1(G, \mathfrak{g})$ the Maurer-Cartan form of $G$. I'm trying to understand why $\displaystyle\frac{d}...
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1answer
24 views

Prove that $\text{exp}(tX)=\alpha_X(t)$ for all $t\in\mathbb{R}$?

Let $G$ be a Lie group and $\mathfrak{g}$ be the lie algebra of $G$. We know that for any $X\in\mathfrak{g}$ there exists an unique $\alpha_X:(\mathbb{R},+)\longrightarrow (G,\cdot)$ one-parameter ...
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61 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 g=...
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40 views

Trace of the product of a Lie algebra and Lie group element

Take $U \in SU(n)$ and $X \in \mathfrak{su}(n)$. What is known about \begin{align} \text{Tr} (UX) \end{align} In particular Are there any useful identities that apply here? When does $\text{Tr} (...
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33 views

What is the rigorous way for a Lie group (SU(n)) element to be “near” another element?

Statement of the problem I'm working with a function $\lambda : SU(n)\times SU(n)\times SU(n) \rightarrow \mathbb{C}$. Given $U_1, U_2, U_3 \in SU(n)$, I'd like to know how to calculate $\lambda (\...
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31 views

references for concrete computations in Lie groups for abstract toplogical concepts

A Lie group is a smooth manifold whose tangent space at its origin is its Lie algebra. Taking an example for lie group such as SL(2), and due to above facts we should then be able to translate the ...
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1answer
26 views

The action of automorphisms on the Riemann sphere

If we are given that the automorphism group of the Riemann sphere is $$Aut\ \mathbb P^1=\{z\mapsto \frac{az+b}{cz+d}:ad-bc=1\}$$ Why this group does not have any proper subgroups that act without ...
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32 views
+50

Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup?

Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts? https://en.wikipedia.org/wiki/Lie_group#...
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1answer
17 views

extending functions from the horizontal bundle to the whole bundle

Let $(M,g)$ be a Riemannian manifold and $G$ a compact Lie group acting freely and isometrically on $M$. Let $\pi \colon M \to M/G$ be the projection to the orbits. Using the metric, we get a ...
2
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32 views

Inner-product on skew-hermitian matrices

Let $$\mathfrak{u}(n)=\{X\in M(n,\Bbb C):X+X^*=0\}$$ where $X^*$ is the conjugate transpose. Then, $\mathfrak{u}(n)$ is a real vector space. Problem. Show that $\langle X,Y\rangle=\...
2
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2answers
51 views

Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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1answer
45 views

In what sense are complex representations of a real Lie algebra and complex representations of the complexified Lie algebra equivalent?

In this book I read Proposition A.1. The irreducible complex representations of a real Lie algebra $\mathfrak{g}$ are in one-to-one correspondence with the irreducible complex-linear ...
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37 views

Reference Request: Lie Theory For Quantum Field Theory

I have encountered the section on non-Abelian gauge theories in Peskin and Schroeder's QFT textbook, and although I am comfortable with the derivation of the Yang-Mills Lagrangian they present, the ...
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47 views

Is the commutator subgroup $[G,G]$ isomorphic to $G/Z_G$?

Let $G$ be a connected reductive Lie group with Lie algebra $\mathfrak{g}$. That means that $\mathfrak{g}=Z_\mathfrak{g}\oplus[\mathfrak{g},\mathfrak{g}]$, and $[\mathfrak{g},\mathfrak{g}]$ is ...
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34 views

Where are these rational functions coming from?

In the proof of the theorem below (Springer, Linear Algebraic Groups), $T$ is a maximal torus of $G$, with dimension $1$, $B$ is a Borel subgroup of $G$ containing $T$, and $U$ is the set of unipotent ...
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102 views

Generators of so(7)

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is isomorphic to $\mathfrak{so}(7)$. Can ...
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1answer
24 views

What is the root system and the Weyl group of the group spin$(2n)$?

Reading on root systems and Weyl groups, unfortunately I am highly confused when it comes to the spin-groups (the two-fold universal cover of SO$(2n, \mathbb{C})$, realizable as a quotiënt in a ...
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1answer
58 views

Cartan decomposition diffeomorphism at the level of (compact) groups

I am trying to understand the Cartan decomposition in the case of a compact group $G$ with subgroup $K$, such that their respective Lie algebras $(\mathfrak{g},\mathfrak{k})$ correspond to a Cartan ...
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1answer
451 views

Map from $SL(2,\mathbb C)/SU(2)$ to $SL(2,\mathbb C)$

I am trying to write a map $s$, such that $s: SL(2,\mathbb{C})/SU(2) \to SL(2,\mathbb C)$. Using Iwasawa decomposition, I see that any element $g$ in $SL(2,\mathbb{C})$ can be written as $g = su$, ...
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13 views

Conjugacy of two elements

I am trying to understand Weyl groups. I read that the Weyl group of the orthogonal group $O(8)$ consists of permutations and an even number of sign changes. Let $T$ be the maximal torus consisting of ...
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1answer
34 views

Reductive homogeneous spaces

If $G$ is a connected Lie group and $K$ is a closed subgroup of $G$ then $G/K$ is a homogeneous space. If $\frak g,k$ are the lie subalgebras of $G,K$ resp. Then under the projection $\pi:G\rightarrow ...
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12 views

About exceptional Lie group E6

How to show the group of determinant preserving linear transformations of $ z $ is isomorphic to $$ \{a \in Isom_\mathbb{C}(z^\mathbb{C},z^\mathbb{C})|det(aX)=det(X),<aX,aY>=<X,Y>\} $$ ...
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34 views

homogeneous dimension of the Heisenberg group [closed]

How to compute the homogeneous dimension of the Heisenberg group $\mathbb C \times \mathbb R $ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ ...
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44 views

Is it possible to study Lie algebras without knowing too much of representation theory?

There's a course on Lie Groups that I'd like to take, but it seems that for various reasons it's a good idea to take Lie algebras along with it. But after having a brief look at the contents of the ...
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1answer
35 views

A noncompact Lie group need not have any nontrivial tori

I don't understand why the following statement is true. 'A noncompact Lie group need not have any nontrivial tori (e.g. $\mathbb{R}^n$). Taking $n=2$, we get $\mathbb{R}^2$. Now consider the set of ...
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8 views

Irreducible tensor for fundamental representation of SU(N=3)

I am trying to calculate the singlets of the tensor product $N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^*$. I know that $N_c \otimes N_c^*=1\oplus (...
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27 views

Introduction to flag manifolds

What is a good self-contained introduction to the geometry of complex flag manifolds?
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1answer
33 views

Using isometric group to describe E7.

I read John C. Baez's paper, The Octonions, and I am wondering the following statement: $$E_7\simeq Isom(\mathbb{(H\otimes O)P}^2).$$ In his contents, I can only figure out $$E_7\hookrightarrow Isom(\...
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1answer
39 views

Riemannian connection on Lie groups

Let $G$ be a Lie group with a bi-invariant metric. Then, the Riemannian connection is given by $\nabla_XY=\frac1 2 [X,Y]$ for all $X,Y\in \mathfrak g$. In the proof: Since $\langle X,Y\rangle$ is ...
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1answer
24 views

Why the induced metric from the lie algebra of lie group $G$ is left invariant.

we say that a Riemannian metric on $G$ is left invariant if $<u,v>_y = <d(L_x)_y u,d(L_x)_y v>_{L_x(y)}$ to introduce a metric on $G$, take any arbitrary inner product $< , >_e$ on ...
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27 views

Partial differential equation transformation

Consider the partial differential equation $$i \frac{\partial \psi}{\partial t} - \left(i\nabla + \mathbf{A} \right)^2 \psi = 0 \tag{1}$$ for the scalar function $\psi(x,y,z)$ and the vector ...
3
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1answer
35 views

The group $\mathrm{Diff}(F)$ and transition functions of a fibre bundle.

Let $M$ and $F$ be differentiable manifolds, and let $F\to E\to M$ be a differentiable fibre bundle over $M$. A trivialising cover $\{(U_i,\phi_i)\,|\,i\in I\}$ of $M$ determines a set $\{t_{ij}:U_{ij}...
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14 views

Equation for plugging in right-invariant vector fields in canonical connection?

Consider a matrix Lie group $G$ with Lie algebra $\frak g$ identified with left-invariant vector fields $\mathcal L(G)$. The $0$-connection is given by: $$ \nabla_{X^l}{Y^l}=\frac{1}{2}[X^l,Y^l]=\frac{...
3
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1answer
50 views

Symmetric decompositions of $SU(2)$ representations.

Let us consider the representation theory of $SU(2)$. There is a unique irreducible representation of dimension $n$ for each $n \ge 1$, which we will denote $\mathbf{n}$, with the defining $2$-...
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28 views

Limit of the commutator of two elements?

Given a Lie group $G$ such the $\mathfrak{g}$ denoted its Lie algebra. Let $[g,g']_{G}$ the commutator of two elements $g,g' \in G$ and denoted by $[X,X']_{\mathfrak{g}}$ the Lie bracket of two ...
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25 views

compact lie group -> real analytic orbits in $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Now, $G$ also acts on $\mathfrak{g}^*$, the dual of the Lie algebra, by the coadjoint-action. My question now is: are ...
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1answer
58 views

Irreducible representations of Heisenberg group

Lately, I've been struggling with the following problem. Let $H$ be the 3 dimensional Heisenberg group and let $\rho:H\to\text{GL}(n,\mathbb{C})$ be a irreducible representation. Show that $n=1$. I ...
2
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59 views

Involutions and Representation of Lie Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$. Is easy to see that ...
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1answer
23 views

what if the infinitesimal generator of a vectorfield vanishes?

Let $(M,g)$ be a riemannian manifold and $H$ a Lie group acting on $M$. Denote by $l \colon H \times M \to M$ and $l_h \colon M \to M$ the action of $H$ on $M$. Now $H$ acts on $TM$ by derivations, ...
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1answer
1k views

Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...
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1answer
66 views

Is the alternating group a lie group

Is the alternating group a lie group. If so what is the lie algebra corresponding to it? This is not a homework questions. I need the dimension of the lie algebra (if one exists) to prove some ...
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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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14 views

Finite dimensional irreducible representations of Sp(2).

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...
1
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1answer
36 views

If G is a compact semisimple Lie group, then its Killing form is negative definite

Theorem: If $G$ is a compact semisimple Lie group, then its Killing form is negative definite. In its proof: Since $G$ is compact there is an Ad-invariant inner product on $\mathfrak g$. Since each ...
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25 views

A quotient by a discrete normal subgroup is locally isomorphic to the group itself

Let $G$ be a connected topological group and let $\Gamma$ be a discrete normal subgroup of $G$. Then why $G$ and $G/\Gamma$ are locally isomorphic?
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27 views

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$?

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$? This question is inspired by noting that if we have a Hamiltonian Lie group action $G\curvearrowright (M,\...
3
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1answer
61 views

Cartan decomposition of SO(2n)

I am trying to understand the Cartan decomposition theory, on the following example : $G=SO(2n)$, and $K=U(n)$, and I'm interested in the manifold $G/K$ (an hermitian symmetric space). 1) How do we ...
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26 views

By which the Heisenberg group is introduced? [closed]

I want to know who was the first who introduced the Heisenberg group and in what year. In the Wikipedia there is just an indication that this group was named in honor of the famous German physicist ...