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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Lie Algebra homomorphism from $\mathfrak{sl}(2,\mathbb{R})$ to $\mathfrak{gl}(m,\mathbb{R})$ is the derivtive of a unique Lie group homomorphism

I want to show that every lie algebra homomorphism $\phi$: $\mathfrak{sl}(2,\mathbb{R}) \rightarrow \mathfrak{gl}(m,\mathbb{R})$ is the derivtive of a unique Lie group homomorphism $\Phi: Sl(2,\mathbb{...
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How can we find or approximate the curve equations of azimuthal and elevation angles between two repeatedly rotating vectors?

Let $f:\mathbb{R}^3\times\mathbb{R}^3\times\mathbb{R}^3\to \mathbb{R}^2$ a function that takes two vectors $\vec{v},\vec{v}_{err}\in\mathbb{R}^3$ and the triple $(\varphi,\theta,\psi)\in\mathbb{R}^3$ ...
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Definition of $Ad(r)$ the automorphism of a Lie-algebra $\mathfrak{g}$ of $G \subset SL(V)$.

Let $G \subset SL(V)$ be a connected algebraic group, acting irreducible on $V$. Consider the Lie-algebra $\mathfrak{g}$ of $G$, which is semisimple and acts irreducible on $V$. Then i want to do the ...
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Showing that the Lie-algebra $\mathfrak{g}$ of $G \subset SL(V)$ is semisimple and irreducible on $V$.

Let $G \subset SL(V)$ be a connected algebraic group, acting irreducible on $V$, where $V$ is a complex vectorspace of dimension $n$ I want to show that the Lie algebra $\mathfrak{g}$ of $G$ is ...
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Zero element in Lie algebra of a Lie group

On page 189 of John Lee’s Introduction to Smooth Manifolds it is stated that the set of all smooth left-invariant vector fields on a Lie group $G$ is a linear subspace of the space of all smooth ...
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Showing $\frac{\partial}{\partial \Phi}\left(\Phi\circ\exp(\varphi)\circ\Phi^{-1}\right) = I - C(\Phi)C(\varphi)C(\Phi)^T$ for 3D rotations

I am trying to reproduce a result from "A Primer on the Differential Calculus of 3D Orientations" - Bloesch, 2016 Consider equations (72) and (73): (72): $\frac{\partial}{\partial \Phi}\left[...
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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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Name for group of conformal linear maps?

A conformal linear map $A : V \to V$ is a map such that $$ \frac{\langle Av, Aw\rangle}{|\langle Av , Aw \rangle |} = \frac{\langle v, w\rangle}{|\langle v, w, \rangle |} $$ for all nonzero vectors $...
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What is the KAK (Cartan) decomposition in $\text{SL}(d, \mathbb R)$ in terms of linear algebra language (and its relation with SVD)?

Let $G= \text{SL}(d,\mathbb R)$ and consider its Cartan decomposition in the Lie group level $G=KAK$. Here $K$ should be the compact group $\text{SO}(d,\mathbb R)$ and $A$ is a diagonal matrix (please ...
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Opposite of the universal covering group

Let $\mathfrak g$ be a finite dimensional real Lie algebra and let $\tilde G$ be the unique simply connected Lie group with Lie algebra $\mathfrak g$. I think that the set $R$ of connected Lie groups $...
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$\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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weight lattice vector which is not a weight of any representation

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 ...
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Which dimension is needed to represent a Lie algebra as a matrix algebra (as in Ado's Thm)?

Suppose $\mathfrak g$ is a real finite dimensional Lie algebra. If I understand it correctly, Ado's Theorem states that there is a real vector space $V$ and an injective Lie algebra homomorphism $\pi:\...
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Commuting generators of different Lie Algebras [closed]

Suppose that I have two different Lie algebras $\mathfrak{a}$ and $\mathfrak{b}$ with generators $M_{i}$ and $N_{j}$. Is it always the case that, the commutator (which is assumed to exist throughout ...
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Is the Maurer-form a 1-form?

I am studying Lie groups and lie algebras following Nakahara book. In 5.6.4 he introduces the concept of Maurer-Cartan one-form in this way: \begin{equation} \theta: X \rightarrow {(L_{g})^{-1}} _{*} ...
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Is there an equivalent condition to $\exp(X+Y) = \exp(X)\exp(Y)$ in an arbitrary Lie group?

I initially wanted to prove that if a Lie group $G$ is abelian, then $\exp(X+Y) = \exp(X)\exp(Y)$ for any $X, Y \in \frak{g}$. I showed that: $$ \frac{d}{dt}(\exp(tX)\exp(tY))= d_e (l_{\exp(tX)\exp(...
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Representation theory of Lie rings

A Lie ring $(M,+)$ is an abelian group with a product $[\ ,\ ]$ (termed as the Lie bracket) satisfying $[x,x]=0$ $[\ ,\ ]$ is bilinear $[[x,y],z]+[[y,z],x]+[[z.x],y]=0,\ \forall\ x,y,z\in M.$ I want ...
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Multiplicity of highest weights and representations

I am wondering whether there is a nice and clean way of detecting the dimension of an irreducible representation of a complex simple Lie algebra simply from the linear expansion of the corresponding ...
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Norm of the adjoint representation and invariant Riemannian distance

Let $G:=\text{SL}(d,\mathbb R)$ and let $Z$ denote the center of $G$ which is a finite cyclic subgroup. Consider a right $G$-invariant Riemannian distance $d$ on the homogeneous space $G/Z$. Let $\...
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Question concerning positive Weyl chamber

I would like to ask for a hint for exercise 22.5 in Bump's book "Lie groups". The setting is as follows: Let $G$ be a (semisimple, connected, simply connected) compact Lie group, choose a ...
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What means describe a Lie Group explicitly?

I have given the Lie goup $G=\{g\in GL(2,\mathbb R)|g^TBg=B\}$ with $B=\begin{pmatrix}1&1\\0&1\end{pmatrix}$. Now I have to describe the Lie group $\frak g$ and then the Lie Algebra $G$ ...
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Understanding Lee J's Proposition 8.41

Hello. The image corresponds to Proposition 8.41 from the book introduction to smooth manifolds by Lee J. There are some statements in the proof that I can't understand at the moment. I put below what ...
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Splitting of Lie groups over codimension 1 subgroups

The following is stated as Theorem 3.1(i) on p. 50f. in Onishchik/Vinberg, Lie groups and Lie algebras III: Let $G$ be a connected real Lie groups. If there is a closed connected normal subgroup $G_1$ ...
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Proving that $\text{GL}(n,\mathbb{R})$ is open at $\text{gl}(n,\mathbb{R})$ and that its tangent space is isomorphic to $\text{gl}(n,\mathbb{R})$

Hello. The image corresponds to chapter 8 of vector fields of the book introduction to the smooth manifolds of the author Lee. Question 1. Why $\text{GL}(n,\mathbb{R})$ is an open set of the vector ...
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The complexification of a compact connected Lie group, is it semi-simple?

I have read that the complexification of a compact Lie group is a reductive Lie group. Question: Let $G$ be a compact connected Lie group and let $G _\mathbb{C}$ be a complexification of $G$. Is $G _\...
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Extension of an involution on $G$ to an involution on $G_\mathbb{C}$

Let $G$ be a compact connected Lie group and $ \sigma :G \rightarrow G $ be an involution on $G$. Let $G^\sigma :=\lbrace g \in G, \sigma(g)=g \rbrace$. Denote by $G_\mathbb{C}$ the complexification ...
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Do $(\mathfrak{g},K)$-modules only deal with real Lie groups?

In Bump's Automorphic Forms and Representations, p. 200, he gives the definition of a $(\mathfrak{g},K)$-module for $\mathfrak{g}=\mathfrak{gl}_n\mathbb{R}$ and $K=O(n)$ being the maximal compact ...
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Lie algebra of general linear group

I’m reading through some Lie algebra notes and I have never studied Lie groups at all but it says that $\mathfrak{gl}_n (\mathbb{F})$ denotes the Lie algebra $(M_n (\mathbb{F}) , [\cdot , \cdot ] )$ ...
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Is any curve in a Lie group the exponential of its tangent vector close to the identity

Say $G$ is a Lie group with identity $e$ and $\gamma:(-1,1) \rightarrow G$ is a smooth curve with $\gamma(0) = e$ and $\gamma'(0)=v$. Intuitively, it seems true that for a very small $\varepsilon>0$...
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Adjoint representation in second tensor power of faithful representation

Let $ G $ be a simple Lie group and $ (\pi,V) $ a faithful finite dimensional representation of $ G $. Consider the action of $ G $ on $ V \otimes V $ by $$ g \cdot (v_1 \otimes v_2)= gv_1 \otimes ...
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1 vote
1 answer
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Adjoint representation as derivative of adjoint map

Question: Let $G$ be a Lie group, how can we associate $$T_e(\text{Aut}(T_eG)) = \text{Der}(T_eG)$$ My motivation for this question is this particular part from my representation theory lecture: We ...
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Runge-Kutta-Munthe-Kaas integration for SE(3)

I am trying to implement in Python the Runge-Kutta-Munthe-Kaas integration for SE(3) for Euler’s method and RK4 for a simple trajectory with constant speed and angular velocity in the body frame. I ...
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2 votes
1 answer
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Proving $\mathbb{R}^3 \textrm{\\} {0}$ not Lie group, without using homotopy equivalence?

From this question, I found that Lie group structure cannot be granted onto $\mathbb{R}^n \textrm{\\} {0}$ for odd n. I am specifically interested in the minimal ...
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3 votes
1 answer
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Why are highest weight modules of integral highest weight $B$-equivariant?

Suppose $G$ is a connected semisimple algebraic group over $\mathbb{C}$, $B \subset G$ is a Borel subgroup, and $T \subset B$ is a maximal torus. Write $\mathfrak g$, $\mathfrak b$ and $\mathfrak h$ ...
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2 votes
1 answer
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Adjoint map $\text{Ad}: G \to GL(\mathfrak{g}), X \mapsto \text{Ad}_X$ is continuous

Let $G$ be a matrix Lie group and $\mathfrak{g}$ the associated Lie algebra. For fixed $X \in G$, define the linear map $\text{Ad}_X : \mathfrak{g} \to \mathfrak{g}$ by $\text{Ad}_X(y) = XyX^{-1}$. ...
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3 answers
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Is the space $\mathbb{PR}^3$ homeomorphic to $\mathbb{PR}^2\times S^1$?

In this Wiki article it is described how the $SO(3)$ is homeomorphic to the projective space $\mathbb{RP}^3$. I would suggest another way which I hope it works. On $S^2$ one may take any direction (...
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Is there a closed-form expression for the exponential map for SO(n), just like how Rodrigues' rotation formula is for SO(3)?

Rodrigues' rotation formula is great since it gives us a faster way to compute the exp() and log() operators for SO(3) compared to the Taylor series formulation. I was wondering if there was a ...
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1 answer
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How does the weyl group act on weights\roots

Let the Weyl group be: $$W=N(T)/T$$ where $T$ is the maximal torus of some lie group $G$ and $N(T)$ is the normalizer of $T$. I saw that in this question that the Weyl group acts on weights by: $$(w.\...
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Strange convergence behavior of non-linear least-squares using Lie algebra for SE(3)

I am having trouble understanding an issue that I observe when running a rather simple bundle adjustment problem using non-linear least-squares with an analytical Jacobian as opposed to a finite-...
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Choosing a norm on a Lie algebra, that is "compatible" with the adjoint representation

I have the following problem: Consider a compact Lie group $G$. Let $\mathfrak{g}$ denote its Lie algebra. We have the exponential map: $$ exp: \mathfrak{g} \rightarrow G $$ It is a fact that we may ...
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2 votes
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Question about finite dimensional representations of a semi-simple Lie group

I've encountered the following paragraph while reading page 3 of this paper https://link.springer.com/article/10.1007/BF01232026?noAccess=true Let $G$ be a semi-simple Lie group with a maximal ...
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6 votes
1 answer
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Is every Zariski-closed real matrix Lie group the joint stabilizer of some list of mixed tensors?

Consider a finite-dimensional real vector space $V$, and an embedded real Lie subgroup $G \subset \mathrm{GL}_\mathbb{R}(V)$. In what follows, $V^*$ denotes the real dual vector space of $V$. Def: Let ...
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2 votes
1 answer
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Which Dynkin diagram is being spoken about here? Why is there a double line?

I'm confused about the following comment in Knapp's Lie Groups 2ed, page 397. Here, $\Delta$ is a root system associated to a complex semisimple Lie algebra, $\alpha, \beta$ are orthogonal roots and ...
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