# Tagged Questions

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

Please show how to solve? I am stack with lie bracket. Thank you.
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### 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: ...
### 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 ...