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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16 views

Can I integrate the Lie-algebra (body angular velocity) of a quaternion?

This is my first mathematics question here. So I am trying to model a 3-d rotation rigid body by Euler's equation. Of course quaternion is the place to go. If in each time step I receive the body-...
4
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1answer
27 views

Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
12
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196 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:...
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23 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 $G$. Let $f_g \colon G \to \mathbb{R}$ be a family of smooth functions $(f_g)_{g \in G}$, is the function $$ G \to \...
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1answer
13 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 $\...
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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
31 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
9 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 ...
9
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1answer
151 views

Are there common inequivalent definitions of Cartan subalgebra of a real Lie algebra?

I'm confused about seemingly different notions of a Cartan subalgebra of a real semisimple Lie algebra, and I'm wondering if there are common inequivalent definitions. In the book Lie Groups: Beyond ...
0
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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 ...
15
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2answers
244 views

Linear Algebra : Invertible Matrix Proof

I was doing some linear algebra exercises and came across the following tough problem : Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose $\...
0
votes
0answers
26 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 \...
0
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1answer
51 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
34 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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48 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
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{...
2
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1answer
30 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 '...
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 $p\in M$ ...
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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. ...
4
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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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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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56 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
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 ...
0
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1answer
41 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} & = \...
8
votes
3answers
852 views

Finding the dimension of the symplectic group

How do you find the dimension of the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$? $\operatorname{Sp}(2n,\mathbb{R})\subset \operatorname{Gl}(2n,\mathbb{R})$ is the group of invertible ...
4
votes
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: \...
1
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1answer
61 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 ...
4
votes
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 $\exp^{-1}$, ...
0
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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 ...
4
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1answer
40 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) ...
4
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1answer
469 views

Proof that $U(n)$ is connected

I'm trying to prove that $U(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I\}$ is connected, but most of the proof comes down to proving that $SU(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I $ and $ detX=1\}$...
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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, ...
3
votes
1answer
516 views

a neighbourhood of identity $U$ generates $G$ where $G$ is a connected lie group

Let $G$ be a connected Lie group and $U$ any neighbourhood of the identity element. How to prove that $U$ generates $G$.
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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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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 ?
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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 $...
3
votes
2answers
158 views

Homogeneous Isotropic Riemannian Manifolds

In John Lee's book Riemannian Manifolds on page 33, Lee writes "Clearly a homogeneous Riemannian manifold that is isotropic at one point is isotropic at every point". It seems that he means that he ...
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1answer
51 views

Isomorphic Lie algebras have isomorphic centers

I think that if two Lie algebras are isomorphic, then their centers should be isomorphic - is this true? I am sure the answer is obvious to those in the know! Here is my attempt at a proof which looks ...
2
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1answer
34 views

What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...
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54 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 $f\in\mathbb{F}[x_1,\ldots,x_n]$ ...
1
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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 ...
0
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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
165 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 ...
4
votes
2answers
4k views

How to evaluate the derivatives of matrix inverse?

Cliff Taubes wrote in his differential geometry book that: We now calculate the directional derivatives of the map $$M\rightarrow M^{-1}$$ Let $\alpha\in M(n,\mathbb{R})$ denote any given matrix. ...
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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
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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 $T$ ...
0
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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}...
0
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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 ...