# Tagged Questions

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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### Finding lie algebra of a group by using exp map and tangent space [duplicate]

I'm studying Lie groups and I am in trouble with finding lie algebras of the classical groups. How can I calculate $\mathfrak{sp}(n,\mathbb{C})$ or $\mathfrak{so}(n,\mathbb{R})$ using exp map and ...
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### Finding the generators of $SU(3)$ different from the Gell-Mann matrices?

I want to find a set of generators of SU(3) different from the Gell-Mann matrices. How should I go about it? Can I construct it in such a way that at least three of the 8 generators when squared gives ...
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### Endomorphism vector bundle isomorphic to the adjoint bundle of its frame bundle?

Could somebody help me to prove the following isomorphism (in particular what is the isomorphism)? End(\xi) \cong ad(E_{\xi}) = E_{\xi} \times_{GL(n,\mathbb{R})} \text{Mat}_n(\...
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### elments of a linear algebraic group agreeing on a vector

Let $G \subset \mathrm{GL}_n(k)$ be a connected affine algebraic group over a field $k$ with the following property: for any two distinct elements $g,h \in G$ there exists a vector $x \in k^n, x\neq 0$...
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### Is the BGG category $\mathcal{O}$ a Serre subcategory of $\mathfrak{g}$-mod? [duplicate]

Let $\mathcal{O}$ be the BGG category for a be a finite-dimensional, semi-simple complex Lie algebra $\mathfrak{g}$. Let $\mathfrak{g}$-mod be the category of all $\mathfrak{g}$-modules. Is the BGG ...
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### Conditions on characteristic polynomial to define a matrix submanifold.

I'm trying to find conditions on the characteristic polynomial, $p$, of a matrix such that the pre-image of matrices with characteristic polynomial $p$ form a manifold. More precisely, we can write ...
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### Existence of Certain Lie Groups

Let $\mathfrak{h}$ be a Lie algebra (not necessarily finite dimensional). Does there necessarily exist a Lie group $G$ such that for the Lie algebra corresponding to $G$, denoted $\mathfrak{g}$, we ...
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### A lie group $G$ is compact iff $G/H$ is.

Suppose that $G$ is compact Lie group and $H$ is closed subgroup. Than $G/H$ is compact since canonical projecton is continuous. Is the inverse true, namely can one prove that: G is compact iff $G/H$...
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### Left Invariant vector field on SO(3)

I have a Lie group, namely on SO(3), i.e. $SO(3,\mathbb{\mathbb{R}})=\left\{ A\in GL\left(3,\mathbb{R}\right)\mid A^{T}A=\mathbb{1},\,\det\left(A\right)=1\right\}$. I have a Left action $L_g$ and I ...
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### Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
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### Showing that the commutator subgroup of a Lie group is a Lie subgroup

I'm learning about Lie groups and Lie algebras independently, and I'm trying to show that the commutator subgroup, $H=[G,G]$, of a Lie group, $G$, is a Lie subgroup. My first instinct was to take a ...
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### Left-invariant vector fields on the circle $S^1$

I'm trying to find the left-invariant vector fields on the circle $S^1$. If I understand correctly, $S^1$ is given the group structure of the multiplicative group of complex numbers on the unit ...
I am reading Steenrod. He writes: Another example of a bundle is a Lie group $B$ operating as a transitive group of transformations on manifold $X$. The projection is defined by selecting a point $... 0answers 29 views ### Topologies of partially exponentiated lie algebras, especially in regard to$SU(2)$Consider the fundamental respresentation of$\mathfrak{su}(2)$given in terms of the Pauli matrices as$\mathfrak{su}(2) = \langle \frac{i\sigma_1}{2},\frac{i\sigma_2}{2},\frac{i\sigma_2}{2}\rangle_{\...
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...