1
vote
1answer
69 views

What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
3
votes
0answers
67 views

Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
7
votes
1answer
138 views

Special conformal killing fields - solving for integral curves.

For each $b\in\mathbb R^d$, let a vector field $X_b:\mathbb R^d\to\mathbb R^d$ be defined as follows: \begin{align} X_b(x) = 2(b\cdot x)x - x^2 b, \end{align} where $x^2 = x\cdot x$. This is the ...
4
votes
1answer
59 views

The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
-2
votes
1answer
39 views

reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
2
votes
0answers
56 views

Gradient of a real-valued function on SO(3)

I have struggling with a problem of evaluating the gradient of a cost function on the Lie group of rotations: SO(3). The cost is the following: \begin{equation} ...
3
votes
0answers
43 views

Actions of Weyl group

I get a feeling what I am going to ask is very standard and classic, but I am not able to find any reference. Any answer or reference would be appreciated. Let us assume that $G$ is a simply ...
1
vote
0answers
32 views

Component of a pushover vector by one-parameter transformation

I am curious about a step on the proof that shows Lie derivative of a vector field is equivalent Lie bracket. Following comes from Nakahara. We define integral curves by vector field X and Y as ...
1
vote
2answers
48 views

Lie algabra of R^n

Until now the only example of lie groups I have seen are subgroups of $GL_n$. Today I had the idea, that also $G=(\mathbb R^n,+)$ must be a lie group ($(\mathbb R^n,+)$ is a group with the ...
3
votes
2answers
52 views

Computing a Lie Bracket: General Questions

I'm asked to compute the following Lie Bracket: $\left [ -y \dfrac{\partial}{\partial x} + x\dfrac{\partial}{\partial y} , \dfrac{\partial}{\partial x} \right] $ on $\mathbb{R}^2$. Just writing it ...
3
votes
1answer
97 views

What is the kernel of a Maurer-Cartan form?

The Maurer-Cartan form on the Lie group $Gl(n,\mathbb{R})$ is a one-form taking values in $\mathfrak{gl}(n,\mathbb{R})$ as defined in the link. It has a rather concrete "extrinsic definition" as ...
0
votes
0answers
63 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
0
votes
1answer
64 views

Tangent space of $\mathfrak{ so}(3)$ Lie algebra

Very basic question and the terminology makes it difficult to find a reference. I just know the basics of differential geometry but my question is simple. Is the tangent space at the point ...
0
votes
1answer
68 views

A question about differential forms on Lie groups

Let $G$ is a Lie group and $\mathfrak{g}$ is its Lie algebra and $\mathfrak{g}^{\mathbb{C}}$ be its complexification. Also assume that $\mathfrak{h}\subset \mathfrak{g}^{\mathbb{C}}$ be its ...
3
votes
0answers
84 views

How was this Lie algebra found?

In a paper the author lists, without justification, generators for a Lie algebra. I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself. ...
1
vote
2answers
137 views

A left invariant vector field on a Lie group

Let $G$ be a matrix Lie group. Let $v$ be a left invariant vector field on $G$ and $v_1 \in \frak g$, where $\frak g$ is a Lie group of $G$. Let $v_1$ be its value at the identity. We define $\phi_t ...
1
vote
1answer
74 views

Lie algebra homomorphism and action on a manifold

In Introduction to smooth manifolds Lee says on page 527: If $\mathfrak{g}$ is an arbitrary finite-dimensional Lie algebra, any Lie algebra homomorphism ...
3
votes
0answers
98 views

Vector fields on smooth manifolds and Lie algebras

I'm currently studying differential geometry on smooth manifolds using differential forms and I'm trying to apply it to what I have learned earlier about Lie groups, but something doesn't seem to ...
1
vote
1answer
54 views

an identity related to moment map

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra, and $X,Y\in \mathfrak{g}$ and also let $\mu:M\to \mathfrak{g^*}$ be moment map($M$ smooth manifold) then prove the following equality ...
0
votes
1answer
53 views

A question about coadjoint orbit

If the coadjoint orbit $\Omega\subset \mathfrak{g^*}$ be contractible then prove that $\Omega$ is integral , i.e., $\int_C \omega\in \mathbb{Z}$ for every integral singular 2-cycle $C$ in $\Omega$, ...
0
votes
0answers
47 views

A $\mathbb{Z}$-graded Lie superalgebra from a Lie algebra

Let $\mathfrak{h}$ be any $\mathbb{K}$-Lie algebra. We set $\mathfrak{g}_{-1}=\mathfrak{h}$ (as vector space), $\mathfrak{g}_0=\mathfrak{h}$ and $\mathfrak{g}_1=\mathbb{K}$ (or any one dimensional ...
2
votes
1answer
49 views

Cartan subalgebra of compact group as “annihilator” of a single element

Let $G$ be a compact Lie group. The Cartan subalgebra $\mathfrak{h}$ can be defined to be a maximal abelian subalgebra of the Lie algebra $\mathfrak{g}$ of $G$. I know this is not the standard ...
4
votes
1answer
128 views

Clifford Algebra for understanding Atiyah Singer Index Theorem Reference Request

I am interested in studying Atiyah Singer Index Theorem and Spin Geometry and would like to study Clifford Algebras and their representations for this purpose. I have a book 'Clifford Algebras : An ...
1
vote
1answer
195 views

Lie algebra of normal subgroup is an ideal

I want to prove that if $G$ is a connected Lie group, $H$ is a normal Lie subgroup of $G$, $\mathfrak{g}$ and $\mathfrak{h}$ their respective lie algebras, then $\mathfrak{h}$ is an ideal of ...
1
vote
1answer
49 views

The tangent space of $\mathrm{Aut}(T_eG)$

Let $G$ be a Lie group and $e \in G$ be the identity. I want to understand the following sentence. " $\mathrm{Aut}(T_eG)$ being just an open subset of the vector space of endomorphisms of $T_eG$, its ...
5
votes
1answer
273 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 ...
1
vote
2answers
77 views

behavior of scalar product defined by trace under commutator

Define $$\langle X,Y \rangle := \operatorname{tr}XY^t,$$ where $X,Y$ are square matrices with real entries and $t$ denotes transpose. I have some troubles in proving that $$ \langle [X,Y],Z \rangle = ...
8
votes
1answer
201 views

Why is the Lie derivative linear in the vector field?

This might seem a very basic question, but I can't manage to find a proper proof in the books I have on my desk (or simply cannot see that it's "just that"). So be sure of what we talk about, let $G$ ...
2
votes
1answer
148 views

left-invariant n-form and metric on a Lie group

These two questions are from my exam practice question sets , which are quite similar. I got some problem understanding and solving both of them . For (a) , I can only substite $dx\wedge dy\wedge ...
2
votes
1answer
89 views

Show that an Ehresmann connection on a principal G bundle is equivalent to a Lie Algebra Valued one form.

Let $E$ be a smooth principal $G$-bundle on M. The vertical bundle $V$ is defined as $V=\ker(d\pi:TE\to \pi^*TM)$. An Ehresmann connection on $E$ is a smooth subbundle $H$ of $TE$ (also called the ...
0
votes
0answers
73 views

Definitions of Semisimple Lie Algebra

We usually define semisimplicity of a Lie algebra $\mathfrak{g}\subset M_n ({\bf R})$ from two ways. I want to know the relation between them. One of the definitions of semisimple Lie algebra is ...
3
votes
1answer
92 views

Compactness of semisimple Lie algebra

I want to prove that on a semisimple Lie algebra $\mathfrak{g}$ over ${\bf R}$: $\mathfrak{g}$ is compact if and only if the Killing form is strictly negative definite. Here the Lie algebra is ...
0
votes
1answer
76 views

Derivations on semisimple Lie algebra

First recall some definitions : Let $B$ be a Killing form on Lie algebra $\mathfrak{g}$ over ${\bf R}$ such that $B(X,Y)\doteq Tr(ad_Xad_Y)$. $\mathfrak{g}$ is semisimple if $B$ is ...
1
vote
1answer
36 views

Lie subalgebra, Lie subgroup and membership

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $H$ be a connected Lie subgroup with Lie algebra $\mathfrak{h}$. We have that $X \in \mathfrak{h} $ iff $exp(tX) \in H \ \ \ \forall t ...
1
vote
0answers
65 views

Proof of Lie theorem on solvable Lie algebra

I am reading a book of Helgason. As you know, solvable Lie algebra $g \subset V= {\bf C}^n$ have a nonzero $v$ such that $v$ is an eigenvector of any element of $g$. I can follow the proof in ...
0
votes
1answer
71 views

Find a $1$-form $ω$ on $\mathbb R^2 −\{(0,0)\}$ such that $ω(X) = 1$ and $ω(Y) = 0$.

Please ı dont know what I need to do. thus, help me to solve.
1
vote
1answer
119 views

Is this distribution involutive?

For two days I've been trying to show the following: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and consider the smooth distribution $$F=\{F_p=DR_p(e)\mathfrak{h}; p\in G\},$$ where ...
1
vote
1answer
84 views

Question about lie bracket..

Let $G$ be a Lie group with Lie algebras $\mathfrak{g}$ and let $\mathfrak{h}\subseteq \mathfrak{g}$ be a Lie subalgebra. Write $F_p=DR_p(e)\mathfrak{h}$, $p\in G$, where $R_p:G\rightarrow G$ given by ...
1
vote
1answer
69 views

Equality involving Lie Brackets

I have a question concerning Lie brackets: Consider the Lie bracket $$[, ]:\mathfrak{g}\times \mathfrak{g}\rightarrow \mathfrak{g},$$ where $\mathfrak{g}=T_eG$ is the Lie algebra of a Lie group $G$. ...
4
votes
1answer
204 views

Lie bracket of vector fields definition equivalence

Lie bracket of vector fields is defined in two ways: Let $\Phi^X_t$ be the flow associated with the vector field $X$, and let $d$ denote the tangent map derivative operator. Then the Lie ...
2
votes
1answer
356 views

Tangent space at the identity element of a lie group

Let G be a lie group . we know a Lie group is a group with a smooth manifold structure s.t both the multiplication map $m$ and group inversion map $i$ are smooth . Now by identifying ...
3
votes
1answer
166 views

Definition of lie bracket of vector fields

The Jacobi-Lie bracket or simply Lie bracket, $[X,Y]$, of two vector fields $X$ and $Y$ is the vector field such that $[X,Y](f) = X(Y(f))-Y(X(f)) \,.$ ...
9
votes
2answers
310 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
2
votes
0answers
83 views

Regarding the definition of vector field flow

To make the connection to the Lie derivative, let $t \mapsto \Phi^X_t$ be the 1-parameter group of diffeomorphisms (or flow) generated by the vector field $ X $. The differential $ d\Phi^X_t ...
5
votes
2answers
66 views

Tangent Space of $\operatorname{Aut}(T_e G)$

Let $G$ be a Lie Group, how to prove that the tangent Space of $\operatorname{Aut}(T_e G)$ at identity is $\operatorname{End}(T_e G)$? Thanks.
6
votes
2answers
615 views

Two Definitions of the Special Orthogonal Lie Algebra

I am encountering two definitions of the special orthogonal lie algebra, and I would like to know if they are equivalent, and if there are advantages to working with one over the other. If we begin ...
5
votes
1answer
266 views

Invariant Inner Product on Lie Algebra

Let $G$ be a Lie group, $\frak{g}$ its Lie algebra. Suppose $\mathcal{D}$ a representation of $G$ on $V$, $d$ the associated Lie algebra representation. Suppose $V$ is endowed with an inner product. ...
4
votes
1answer
97 views

The set of complete vector fields

The set of all complete vector fields in $\mathbb R^{n}$ is closed under Lie bracket? is this set a $D$-module where $D$ is the ring of bounded smooth funcions? Can anyone recomend me a book on the ...
3
votes
1answer
108 views

Induced metric via $\mathbb C P^n \cong SU(n + 1)/S(U(n) + U(1))$

I was wondering if the homeomorphism above gives me the Fubini Study metric on $\mathbb C P^n$. More precisely: Consider $\mathbb C P^n$ equipped with the metric induced by the standard construction ...
1
vote
0answers
204 views

jacobian involving SO(3) exponential map: $\log(R * \exp(m))$

I would like to compute the 3x3 Jacobian of $$ \log(R * \exp(m)) $$ with respect to the 3-vector $m$, evaluated at $m=0$. In the above, $\exp$ is the exponential map from so(3) to SO(3), $\log$ is ...