# Questions tagged [lie-groups]

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-theory) tag.

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### Recovering the definition of exponential matrix from the abstract definition of Lie groups.

I am studying the exponential function of the book introduction to the smooth manifold by John Lee and the following question has arisen. Let $\exp:\mathcal{G}\to G$ exponential map, with $G$ a Lie ...
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### Inertia subgroup of a p-adic Lie group is infinite

It is stated here (very end of the 3rd page) that the inertia subgroup of the p-adic Lie group ${\rm Gal}(K_{\infty}/K)$ is infinite when the dimension of ${\rm Gal}(K_{\infty}/K)$ is greater than $2$ ...
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### Time derivative of the blend of a pair of quaternion curves

I have two curves ${\bf q}_0(t), {\bf q}_1(t)$. Each curve maps time $t$ to a unit quaternion. Construction of these curves is not important here, although we do have the respective time derivatives ...
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### $\exp$ of a Lie group is a local diffeomorphism at all points of the Lie algebra

I am trying to show that for any Lie group $G$ with $T_e G=:\mathfrak{g}$, $\exp:\mathfrak{g} \rightarrow G$ satisfies that $d_X \exp : T_X \mathfrak{g} \rightarrow T_{\exp(X)}G$ is invertible for any ...
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### What are the implicit conditions underlying the $J$ matrices for $J$-orthogonal matrices?

A square matrix $Q$ is said to be $J$-orthogonal if $$Q^T J Q = J.$$ What are the implicit conditions on this matrix $J$? I believe $J$ is typically chosen to be a symmetric non-degenerate form, but ...
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### Can we read the quanternion in terms of eigenvalues of equivalent 3D rotation matrix?

What is the relationship between a quanternion and the eigenvalues and eigenvectors of equivalent 3D rotation matrix? A relation in terms of $\lambda_r, \mathbf{v}_{\lambda_r}, \mathbf{v}_{\lambda_1}$ ...
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### Induced maps on homotopy groups of $SU(2) \rightarrow G$

We have $\pi_3(G) = \mathbb{Z}$ for all compact connected simple Lie groups, and we know that given a map $\phi: SU(2) \rightarrow G$, the induced map $\phi_{*} : \mathbb{Z} \rightarrow \mathbb{Z}$ on ...
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### Equivalence between bi-invariant metrics on Lie groups and Symmetric spaces

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and $K$ a connected closed Lie subgroup of $G$ with Lie algebra $\mathfrak{s}$. Then $G/K$ is a homogeneous space. Equip $G$ ...
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### (invariant) Distance from geodesic flow matrix to the identity in $\text{SL}(m+n,\mathbb R)$

Let $\text{SL}(m+n,\mathbb R)$ be equipped with a left or right invariant Riemannian distance $d$ (this is part of the question). Let $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$ where $I_m, I_n$ denote ...
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What is an example of a simply connected Lie group $G$ and an integral vector $\lambda$ (i.e. a vector in the weight lattice) such that $\lambda$ is not a weight of any finite dimensional ...