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 ...
6
votes
1answer
82 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
1
vote
1answer
75 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 ...
0
votes
0answers
21 views

Abelian Lie Group [duplicate]

Take a Lie group G and consider the tangent space at one of its points. In particular, the tangent space at the identity e is usually denoted by g := $T_e G$. Can you prove that, if G is an abelian ...
4
votes
1answer
62 views

Exterior Derivative Problem

Suppose $\theta$ is a differential $1$-form defined on a manifold and with values in the Lie algebra of a Lie group $G$. On $M\times G$ define the $1$-form $ad(g)\theta$ where $\theta$ is extended ...
1
vote
1answer
56 views

if $X$ is a vector field how can I find $Y$ such that $[X,Y]=0$?

Suppose I am given a holomorphic vector field $X$ over a complex manifold $M$. To simplify this we can suppose that $X$ is a holomorphic vector field in $\mathbb{C}^n$ for $n=2$ or $n=3$. How can I ...
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 ...
2
votes
1answer
357 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 ...
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 ...
0
votes
1answer
67 views

Lie bracket of vector fields on $\Bbb R^{n}$

Please show how to solve? I am stack with lie bracket. Thank you.
6
votes
3answers
393 views

Lie algebra action from Lie group action: coordinates

Here's the setup: I have $SL(2;\mathbb{C})$ acting on $V = \mathbb{C}[z,w] = \oplus_d V_d$, where $V_d$ is the homogeneous complex polynomials of degree $d$. The action is precomposition: ...
2
votes
1answer
260 views

Two Lie algebras associated to $GL(n,\mathbb{C})$

I have elementary questions about Lie groups and their associated Lie algebras. Let $G=GL(n,\mathbb{C})$. Then associated to this Lie group is the Lie algebra $M_n(\mathbb{C})$ with the commutator ...
2
votes
1answer
204 views

Conditions for left-invariant one-forms to be closed.

Let $G$ be a connected (semisimple) Lie group with Lie algebra $\frak{g}$. For $\omega \in \frak{g}^*$, we may define a left invariant one-form on $G$ by $\left[ \omega (g)\right] (v)=\omega \left( ...
3
votes
2answers
131 views

A certain subset of $\mathfrak{u}(n)$ is an embedded manifold?

I would have a hint on how to control if the following subset $C$ is an embedded submanifold of $\mathfrak{u}(n)$. $C$ is the set of the antihermitian $n\times n$ matrices $A$ with the property that ...