Tagged Questions

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20 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 ...
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1answer
20 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 ...
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1answer
25 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 ...
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0answers
31 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 ...
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1answer
60 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.
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1answer
36 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 ...
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1answer
58 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 ...
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1answer
53 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 $G$. ...
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0answers
68 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
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1answer
57 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 ...
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1answer
57 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 $[X,Y](f) = X(Y(f))-Y(X(f)) \,.$ ...
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2answers
107 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
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0answers
35 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 ...
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2answers
58 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.
3
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1answer
98 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 ...
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0answers
58 views

Proof that inversion map ($g \mapsto g^{-1}$) of a Lie groupoid is diffeomorphism

I am now reading "General theory of Lie groupoid and Lie algebroids" by Kirill C. H. Mackenzie. I am going over the proof of the following proposition (p. 6): Proposition 1.1.5. Let $G$ be a Lie ...
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1answer
103 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. ...
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0answers
55 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 ...
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1answer
54 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 ...
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0answers
69 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 ...
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2answers
162 views

On surjectivity of exponential map for Lie groups

A recent question made me realize I didn't know any proof that exponential of a Lie algebra $\mathfrak g$ of a compact connected Lie group $G$ is surjective. After a bit of thinking I've come up with ...
3
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2answers
61 views

Nilpotent Lie Group that is not simply connect nor product of Lie Groups?

I have been trying to find for days an non-abelian nilpotent Lie Group that is not simply connect nor product of Lie Groups, but haven't been able to succeed. Is there an example of this, or hints to ...
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1answer
93 views

Lie bracket and connection of a surface

Let $f:U \to \mathbb{R}^3$ be a surface where $U \subset \mathbb{R}^2$ is open.Let $\Gamma(Tf)$ denote the space of smooth tangent vector fields on $f$ A connection on $f$ is a map $D:\Gamma(Tf) ...
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1answer
298 views

Lie derivative: concrete example for linear Lie group

I am trying to understand the notion (and notation) of the Lie derivative on a general manifold by trying to convert the notation the concrete example of the Lie group O(n). Let $X,Y$ be smooth ...
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1answer
102 views

kernel of adjoint of Lie algebra

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. The adjoint representation of the Lie algebra $\mathfrak{g}$ is defined as: $$ \text{ad: } \mathfrak{g} \rightarrow ...
2
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1answer
87 views

Lie algebra of a Lie subgroup

Let $G$ be a Lie group and $H$ a Lie subgroup of $G$, i.e. a subgroup in the group theoretic sense and an immersive submanifold. Let $\mathfrak{g}$ and $\mathfrak{h}$ be the associated Lie algebras. ...
2
votes
1answer
174 views

left-invariant vector field: counterexample

Let $G$ be a Lie group, $L_g$ the left-translation on this group with differential $d L_g$. A vector field $X$ on $G$ is called left-invariant if $$ X \circ L_g = d L_g \circ X \quad \forall g \in ...
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3answers
362 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 ...
3
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1answer
119 views

Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?

Let $G$ be a (EDIT: connected) Lie group of dimension $n$, and let $\mathfrak{g}$ be the associated Lie algebra. If $x_1,\ldots,x_n$ is a basis for $\mathfrak{g},$ is it necessarily true that the ...
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0answers
95 views

Derivative/Chain Rule (for MANLYfolds) Computation

Embarrasingly, I can't compute the following derivative. $dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}$, where $X$ resides in the lie algebra of $\rm SL(3,\Bbb C)$ [ie $\mathfrak{sl}(3,\Bbb C)$] ...
3
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1answer
237 views

Lie Algebra Homomorphism Question

So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
3
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0answers
82 views

The Nambu bracket

Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $$\{F,G\}_c = \langle\nabla ...
1
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1answer
610 views

Lie derivative of a vector field equals the lie bracket

Let $X$ and $Y$ be vector fields on a smooth manifold $M$, and let $\phi_t$ be the flow of $X$, i.e. $\frac{d}{dt} \phi_t(p) = X_p$. I am trying to prove the following formula: $\frac{d}{dt} ...
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4answers
377 views

What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
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0answers
166 views

$SU(2)$ is a covering space of $SO(3)$.

The method of topology is very clear.Then there's a question asking to use adjoint representation of lie group $SU(2)$ $(\operatorname{adj}:SU(2)\to GL(su(2)))$to prove this. I can't solve this .
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0answers
108 views

First-order derivatives in differential forms calculus

Let $d$ denote the Cartan differential, and let $\delta$ denote the codifferential. The underlying domain is not important for what follows. The canonical generalization of the Laplace-operator ...
4
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0answers
122 views

internal direct product of lie groups

If $G$ is a (edit: simply connected)Lie group, when does a direct sum decomposition of its Lie algebra (into a direct sum of subalgebras) correspond to a (semi)direct product decomposition of $G$? ...
2
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0answers
128 views

Lie Group Multiplication in Coordinates

I'm having a bit of trouble with the last bit of Problem 3.2 in Kirillov Jr.'s Introduction to Lie Groups and Lie Algebras. (3.2) Let $f: \mathfrak{g} \rightarrow G$ be any smooth map such that ...
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1answer
216 views

Definition of tangent space

Terry Tao defines tangent space here as equivalence classes of continuously differentiable curves $\gamma : I \rightarrow G$ where $I$ is an open interval. On the other hand, Wikipedia defines it as ...
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1answer
110 views

Relation between a Lie group and Lie algebra representation for $W \otimes V$

We can define a representation of a Lie group and get the induced representation of the Lie algebra. Let $G$ act on $V$ and $W$, $\mathfrak{g}$ be the Lie algebra associated to $G$ and $X \in ...
4
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0answers
246 views

Elementary proof of the third Lie theorem

I would like to understand a (not well known) proof due to G.M; Tuynman, of the third Lie theorem, which asserts that for any given finite dimensional Lie algebra $\mathcal{G}$ there exists a (simply ...
0
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1answer
300 views

What is meant by adjoint of a linear transformation w.r.t a given inner product?

Consider a matrix Lie group equipped with a left &/or right invariant metric. The adjoint of linear transformation $A$ with respect to the inner product is denoted as $A^*$. Here what is ...
2
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1answer
884 views

Classsifying 1- and 2- dimensional Algebras, up to Isomorphism

I am trying to find all 1- or 2- dimensional Lie Algebras "a" up to isomorphism. This is what I have so far: If a is 1-dimensional, then every vector (and therefore every tangent vector field) is ...
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1answer
304 views

Relationship between Riemannian Exponential Map and Lie Exponential Map

It is well known that for a matrix Lie group the Lie exponential map is $e ^z$. This maps a tangent vector $z$ at the identity to a group element. On the other hand the general Riemannian ...
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2answers
223 views

Books on Lie Groups via nonstandard analysis?

Are there any books or online notes that cover the basics of lie groups using nonstandard analysis? Another thing I would like is a to see these things set in category theory (along the lines of ...