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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8 views

Conjugacy of Cartan subalgebras

This is probabably a very silly question, stemming from some fundamental misunderstanding I have of the relevant definitions, but I am stumped by it. I know that any two Cartan subalgebras of $\...
2
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0answers
12 views

Integration over the Haar measure of a compact Lie group preserves smoothness?

Let $G$ be a compact Lie group. Then there is a unique Haar (probability) measure on $G$. If $f \colon G \to \mathbb{R}$ is a smooth function, is the function $$ G \to \mathbb{R}, \qquad x \mapsto \...
0
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0answers
17 views

Different ways of decomposing an exponential map

There are many decompositions of an exponential map which has two (or more) operators in the exponent (i.e. $e^{A+B}$, where $A$ and $B$ are operators). For example, the Baker-Campbell-Hausdorff (and ...
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1answer
30 views

$\pi_0(SO(N))$ and $\pi_0(O(N))$: Inconsistency between Bott periodicity and basic understanding of $\pi_0$

I need to know the homotopy groups of the oriented Grassmannian $\widetilde{Gr}(\infty,\infty) \cong \lim_{N \rightarrow \infty} SO(2N)/(SO(N) \times SO(N))$, and I've run into an inconsistency. It ...
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0answers
8 views

Multiple of roots in symmetric spaces

Fix a Cartan subalgebra $\mathfrak{h}$ on a (compact simple) Lie algebra $\mathfrak{g}$ and consider the associated root system. If $\alpha$ is a root, it is well-known that $k\alpha$ is also a root ...
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17 views

Explicit matrix representation of $\mathfrak{sl}_3$

Given a semisimple Lie algebra $\mathfrak{g}$ and a dominant integral weight $\lambda$ (and all the other necessary data), I want to be able to write down a matrix representation for $V(\lambda)$, the ...
0
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0answers
8 views

compact Lie group with non-compact Lie subgroup? [duplicate]

Can there be compact Lie groups with non-compact subgroups? I thought that was not possible until I thought of the torus with the irrational rotations. So if one identifies $U(1)\times U(1)$ with the ...
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24 views

Commutators in the context of local Lie groups.

Let $G$ be a local Lie group in the neighbourhood $V \subseteq \mathbb{C}^d$ with identity element denoted by $e \in G$. Also, let $$ t \mapsto f(t) = (f_1(t), \dots, f_d(t)) \quad \forall t \in \...
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1answer
48 views

What is the determinant of exp(matrix)? [duplicate]

Given a square matrix $A$, form the Lie series of it, which is defined by: $$ e^A = I + A + \frac{1}{2} A^2 + \frac{1}{3!} A^3 + \cdots + \frac{1}{n!} A^n = \sum_{k=0}^\infty \frac{1}{k!} A^k $$ Is ...
2
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0answers
33 views

Deriving an Element of the Lorentz Group SO(1, 3)

We know that $SO(1, 3)$ is isomorphic to $SU(2) \otimes SU(2)$: $$SO(1, 3) \cong SU(2) \otimes SU(2)$$ We also know that $$ \left(\frac{1}{2}, \frac{1}{2}\right) = \left(\frac{1}{2}, 0\right) \...
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1answer
39 views

Can we lift paths of a Lie group quotient $G\to G/H$?

Question: Let $G$ be a Lie group and $H\subseteq G$ a closed normal subgroup. Let $$\pi:G\to G/H$$ be the quotient map. If $\gamma:[0,1]\to G/H$ is a smooth path, can we find a smooth path $\tilde{...
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0answers
6 views

Minimize the inner product of this tensor function

Minimize the following function: $ f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product. ...
3
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1answer
56 views

How do we call a map $F$ such that $F(g\cdot p)=\varphi(g)\cdot F(p)$?

Let $G$ and $H$ be groups acting on sets $M$ and $N$. Suppose that there is a group homomorphism $\varphi:G\to H$ and a map $F:M\to N$ such that $$F(g\cdot p)=\varphi(g)\cdot F(p)$$ for all $p\in M$ ...
2
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1answer
28 views

Irreducible representations of the fundamental group of a closed surface in $SU(2)$

For a compact Lie group $G$, consider the map $f : G^{2n} \to G$ given by $f(A_1, B_1, \ldots, A_n, B_n) = \displaystyle\prod_{i = 1}^{n} A_i B_i A_i^{-1} B_i^{-1}$ A theorem of Goldman (from the '...
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0answers
19 views

Constructing element of the Weyl Group

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{h}$ a CSA with root system $\Phi$, base $\Delta$, and Weyl group $W$. Then there exists a unique element $\sigma\in W$ such that $\sigma(\...
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0answers
45 views

Show that Heisenberg group is homeomorphic to $R^3$ as topological space. What is that homeomorphism? [closed]

Show that Heisenberg group H is homeomorphic to $R^3$ as topological space. Is Heisenberg group semi-simple?
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1answer
34 views

Why is SU(2) not the same group as T3?

Why is $S^3 = SU(2)$ not the same group as $ S^1 \times S^1 \times S^1 $? It seems that, the $T^3$ torus is abelian, wheras $S^3$ is not. Is that enough? Problem 2.6.5 from John Stillwell, Naive ...
8
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0answers
54 views

Finite subgroup of $\text{SO}(3)$ acts on set of points on unit sphere in $\mathbb{R}^3$ which are fixed via some nontrivial rotation in $G$

Let $G$ be a finite nontrivial subgroup of $\text{SO}(3)$. Let $X$ be the set of points on the unit sphere in $\mathbb{R}^3$ which are fixed by some nontrivial rotation in $G$. I have two questions. ...
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1answer
40 views

2nd order derivative of Lie group SO(3)

In P.4 of this technical report there is a equation: \begin{align} \left.\frac{\partial^{2}}{\partial \omega_{x}\partial\omega_{y}}(\mathbf{R}_{0}\exp\{J(\omega)\}) \right|_{\omega=0} & = \...
4
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1answer
63 views

Integral formula for the differential of matrix exponential

This is a problem from Jacques Faraut's Analysis on Lie Groups. Given $A,X\in M(n,\mathbb{R})$, put $F(t)=\exp(t(A+X))$. In the first part of the problem we showed that $F$ is a solution to the ...
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2answers
57 views

Munkres Problem: Define a $C^{\infty}$ map $f: \mathbb{R}^9 \to \mathbb{R}^6$ such that $O(3)$ is the solution of $f(x)=0$.

On Munkres's book analysis on manifold chap "the boundary of manifold", question 3, says: let $O(3)$ the set of orthogonal matrices, as a subspace of $\mathbb{R}^9$. a) define a $C^{\infty}$ map $f: \...
4
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2answers
105 views

Why are Lie groups automatically analytic manifolds?

In the book by Kolar, Michor, and Slovak, it is shown that multiplication $\mu:G\times G\to G$ is analytic in some neighborhood of $e$. Specifically, they show that in the chart given by $\exp^{-1}$, ...
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1answer
26 views

Is the connected component of the center of a group equal to the connected component of the center of its solvable radical?

Given a linear algebraic group, is the connected component of the identity of its center equal to the connected component of the identity of the center of its solvable radical? If not, is there a ...
12
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190 views

Derivation of a representation through a vector field

Question: (Exercise 3.4.12 - Sharpe) Let $H$ be a Lie group, $V$ a vector space, and $\rho: H \to Gl(V)$ a representation. Let $U$ be a manifold, $X$ a vector field on $U$, and $h: U \to H$ and $f:...
1
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1answer
60 views

Maurer Cartan Form of the Heisenberg group

I'm trying to understand meaning and application of the Maurer Cartan Form, but I'm still not quite there. I'm then trying to do some examples and trying understand how it works. I begun with the ...
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votes
0answers
14 views

infinitesimal generators of the conformal inversions

More broadly speaking, this question involves the subgroups of the conformal Lie groups on Euclidean space. With some insight, one will know that these consist of the infinitesimal rotations, ...
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1answer
27 views

Regular Representation of Lie Algebras

I have a basic understanding of Lie Algebra and it may be naive but is there a regular representation of lie algebras as in case of Finite Groups ? Do the generators form a representation ?
4
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1answer
39 views

Complex representation of a quaternionic matrix

It is evident that right module $\mathbb{H}^n$ is $\mathbb{C}$-linearly isomorphic to $\mathbb{C^{2n}}$ with corresponding isomorphism $\nu : \mathbb{C^{2n}} \to\mathbb{H}^n $ given by $ \nu(a,b) ...
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0answers
10 views

Cocompact Lattice in $SL(n,\mathbb{R})$.

Can someone give an explicit description for a cocompact lattice in $SL(n,\mathbb{R})$, for $n\geq 3$? (By a cocompact lattice we mean a discrete subgroup $\Lambda$ of $SL(n,\mathbb{R})$ such that $...
0
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1answer
26 views

Is the subgroup of homotopically trivial isometries a closed subgroup of the isometry group?

Let $(M,g)$ be a connected Riemannian manifold. Then according to the Steenrod-Myers-Theorem, the isometry group $\text{Isom}(M,g)$ of $(M,g)$ is a compact lie group with the compact-open topology. ...
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53 views

Lie algebra equivalent definition

I am reading the paper "Affine projections of polynomials" by Neeraj Kayal. I need a clarification regarding the equivalence of two definitions of lie algebra : Let $f\in\mathbb{F}[x_1,\ldots,x_n]$ ...
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3answers
52 views

Faithful representation of the Heisenberg group

I have been trying to solve a problem concerning the Heisenberg Lie group $H$. Show that there does not exist a faithful representation $\rho:H\to\text{GL}(2,\mathbb{R})$. Any ideas about how to ...
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0answers
33 views

$SL_2(R)/SL_2(Z)$ non-compact

Why is $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$ non-compact? I have a hit that says: How $SL_2(\mathbb{Z})$ and $SL_2(\mathbb{R})$ act on $\mathbb{Z}^2-\{0\}$, but I can not understand the hint.
6
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2answers
164 views

understanding relevance of Lie vs topological groups

A silly easy to state question. When dealing with topological groups, I'm trying to understand more profoundly the advantages of having a Lie group structure against just a topological one. Can ...
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0answers
20 views

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 ...
0
votes
1answer
15 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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1answer
28 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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0answers
37 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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0answers
34 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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0answers
32 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}...
1
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1answer
28 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 ...
1
vote
1answer
18 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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0answers
34 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=\...
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0answers
39 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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0answers
43 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} (...
2
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0answers
50 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 ...
0
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1answer
36 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 ...
6
votes
0answers
134 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 ...
1
vote
1answer
25 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 ...
2
votes
1answer
35 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 ...