# 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 ...

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### How to prove that the adjoint group is a Lie subgroup of $Gl(\mathfrak{g})$

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $Ad: G \rightarrow GL(\mathfrak{g})$ be the Adjoint representation. I want to prove that $Ad(G)$ is a Lie subgroup of $GL(\mathfrak{g})$. ...
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### Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $\mathbf{R}^3$ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
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### Clarification of notion of proper group action.

In a course on differential manifolds and Lie groups, the following theorem was stated, though never proven: Let $M$ and $N$ be smooth manifolds, and suppose $G$ is a Lie group acting on $M$. If ...
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### Tangent space to lie group at identity.

I'm supposed to show that for a Lie group G, $T_{(e,e)}G\times G \simeq T_eG\oplus T_eG$ and that $T_{(e,e)}m$ is given by $(X,Y)\mapsto X+Y$. I'm having trouble proving this. I'm not exactly clear ...
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### Straight forward derivation of the bch formula?

Im doing a project on rigid body dynamics and need to derive the bch formula, anyone know a simple yet complete derivation?
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### Adjoint representation of the isotropie group of a homogeneous space

I have difficulties seeing why is the following true: Let $G$ be a lie group and $H$ a closed subgroup, with $\tilde{g}$ and $\tilde{h}$ their lie algebras. The adjoint action of $g\in G$ is given by ...
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### Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
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### Exponentials of Representations of Lie Algebras

Assume G is a lie group and g is its lie algebra. Consider a representation of G : D:G->End(V). Then there is a corresponding representation of g : d:g->End(V). My question is, when you can express ...
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### Compactness of Lie groups

Let $G$ be a Zariski-closed subgroup of $GL(V)$, where $V$ is an $n$-dimensional complex vector space. Question. Does $G$ have the structure of a compact Lie group? Such $G$ certainly is a Lie ...
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### Are the orbits of a connected Lie group acting on a vector space always embedded manifolds?

Setting: We have a connected Lie group $G$ and a smooth map $G \to GL(V)$, where $V$ is a finite-dimensional vector space. Are the orbits of $G$ on $V$ embedded submanifolds? More precisely, if one ...
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### classifying map of associated bundles

Let $G\leq GL(\mathbb{R}^n)$ be a group and $\xi$ be a principal $G$-bundle over a space $X$. Let $\eta=\xi[\mathbb{R}^n]$ be the associated vector bundle of $\xi$. Let $f_\xi: X\to BG$ be the ...
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### Geodesic completeness of a Lie group

Let $G$ be a Lie group and $\rho$ some left(right, bi)-invariant Riemannian metric on $G$. Is it possible to say for which $\rho$ an underlying manifold $G$ is geodesically complete (maybe for every ...
I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Where $\phi(t)$ is a one-parameter subgroup of the Lie group ...