For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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42 views

The derived algebra is a Lie subalgebra

A (hopefully) very simple question that has been bugging me all day! Let $L$ be a Lie algebra then the derived Lie algebra $L'$ is $$ L' = \{ \, [u,v] : \forall u,v\in L \, \}. $$ I want to show ...
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69 views

Computing Jacobian of error function using Lie Algebra

First off all, I hope this is the right place to ask, as it is a computer vision problem, but I'm specifically asking about the mathematical part of it. I am currently implementing the ICP (Iterative ...
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0answers
25 views

coboundary operators in relative lie algebra cohomology

I am starting to read relative lie algebra cohomology. We define the coboundary operator $d$ from $Hom_K(\wedge^q\mathcal{g}/\mathcal{k}, V)$ to $Hom_K(\wedge^{q+1}\mathcal{g}/\mathcal{k}, V)$ as ...
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1answer
41 views

Prove that the tangent space to group of unipotent matrices is a subspace of M(2,R).

Given the set of unipotent matrices: $S = \left\{ A\in GL_{2}(\mathbb{R}) \;:\; A=\left( \begin{matrix} 1 & a \\ 0 & 1 \end{matrix} ...
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56 views

Borel subalgebras inside the grassmannian

This is probably something standard and I just don't know where to look (so a reference would be just as appreciated as an answer), but... Let $\mathfrak{g}$ be a finite dimensional semisimple Lie ...
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46 views

How to tackle a research journal - level course in Lie Theory and Representation Theory?

I am taking a course in Lie Theory and Theory of Representations this year, where starting from the second lecture, Lie Theory is heavily bundled with Theory of Representations. It is pretty much a ...
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80 views

Can any one recommend a way to “quickly” learn a subject?

I would love to read a well written book on a subject - provided that I have the time. But sometimes we do not need to become experts on a particular field but still need the basics. For example, a ...
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56 views

(Split) Exact Sequence of Lie Algebra Associated to Groups

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...
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1answer
128 views

Meaning of the adjoint representation of a Lie group

The adjoint representaion of $G$ is a homomorphism $ad_{a}:g \rightarrow aga^{-1}$, $a,g \in G$, what is the meaning of this? Now if we identify $T_{e}G$ with $\mathfrak{g}$ we have the adjoint map ...
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1answer
84 views

Are the weights of an irreducible representation of a simple Lie algebra in a single Weyl orbit?

When we consider the weights of an irrep of a simple Lie algebra, are they always in a single orbit under the Weyl group of the Lie algebra, or do they form a set of disjoint orbits? If they form ...
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235 views

Lie groups, Lie algebra and left invariant vector fields

Hi I'm learning about Lie Groups to understand gauge theory (in the principal bundle context) and I'm having trouble with some concepts. Now let $a$ and $g$ be elements of a Lie group $G$, the left ...
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71 views

Show that the only invariant is the spectrum

Recall that the symplectic group $$Sp_2(\mathbb{R}):= \{A\in SL_2(\mathbb{R}):A^TJA=J\}, \ \ J= \left[ {\begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array} } \right] \ $$ We have its ...
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1answer
38 views

The Weyl group of $\widehat{\mathfrak{sl}}_2$.

On page 5 of this paper, example 3.1, it is said that the Weyl group of $\widehat{\mathfrak{sl}}_2$ is $$ W= \langle s_1, s_2 \mid s_1^2 = s_2^2 = 1 \rangle. $$ Why the Weyl group of ...
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1answer
109 views

One parameter subgroup

I am new to Lie group and I am reading the "Lie Groups, Lie Algebras, and Representations" by Brian Hall. So what's the intuitive idea about one parameter subgroup? I understand all the definition but ...
2
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1answer
41 views

Decomposing $\mathfrak{sl}_3(\mathbb{C})$

There is a pretty standard exercise on $\mathfrak{sl}_2 (\mathbb{C}$) representations that consists in decomposing the representation given by $\mathfrak{sl}_3(\mathbb{C})$ via $\operatorname{ad}$, ...
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124 views

John Lee's book question about symplectic group

Exercise 12-14 in John Lee's book, An introduction to Smooth Manifolds, reads as follows: The real symplectic group is the subgroup $Sp(n, \mathbb{R}) \subset GL(2n, \mathbb{R})$ consisting of ...
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58 views

Lie algebra associated to an arbitrary discrete group

I read somewhere that there is a classical (due to Philip Hall?) construction of a Lie algebra associated to any discrete group $\pi$ which is obtained from filtration quotients of the descending ...
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2answers
360 views

Meaning of Exponential map

I've been studying differential geometry using Do Carmo's book. There's the notion of exponential map, but I don't understand why it is called "exponential" map. How does it has something to do with ...
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1answer
34 views

Lie group as a subset of its Lie algebra

Consider a (possibly infinite-dimensional) Lie group $\mathcal{G}$ and let $\mathcal{A}$ be an algebra with a product $\cdot$ and the bracket $[u,v]=u\cdot v - v\cdot u$. The following statement is ...
3
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2answers
159 views

A compact connected solvable Lie group is a torus

I am looking for the proof of the following statement. A compact connected solvable Lie group of dimension $n\geq 1$ is a torus, i.e., it isomorphic to the product of $n$ copies of $S^1$. A ...
9
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1answer
405 views

Campbell-Baker-Hausdorff formula for $\log(\exp(X+Y)\exp(X-Y))$

Given $X,Y\in \mathfrak g\mathfrak l_{\mathbb R}(n)$, and the CBH formula for $\log(\exp X\exp Y)$ (wiki), is it possible to derive the general term in the series of $\log(\exp(X+Y)\exp(X-Y))$ that ...
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42 views

Finite-dimensional, irreducible Representations of the Diffeomorphism Group $Diff(R^4)$

Is there any possible way to study the finite-dimensional, irreducible representations of $Diff(R^4)$ systematically? My interests stems from the fact, that the symmetry group of general relativity is ...
4
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1answer
103 views

Lie algebras of vector fields over $\mathbb{R}$ and over $S^1$

An exercise in the book "Moonshine beyond the Monster" has me stumped. It asks whether the real Lie algebras of smooth vector fields over the reals $V(\mathbb{R})$ and over the circle $V(S^1)$ are ...
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1answer
33 views

$Hom_G(\pi,\sigma)$ = $Hom_{\mathfrak{g}}(d\pi,d\sigma)$?

Let $G$ be a Lie group. Let $\mathfrak{g}$ be the corresponding Lie algebra. Let $(\pi,V)$ and $(\sigma, W)$ be representations of $G$, with corresponding differentials $d\pi$ and $d\sigma$, which are ...
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67 views

What is $\Delta(1)$ for $1$ in $U(\mathfrak{g})$?

Let $\mathfrak{g}$ be a semisimple Lie algebra and $U(\mathfrak{g})$ its universal enveloping algebra. Then $U(\mathfrak{g})$ is a hopf algebra. Is $\Delta(1) = 1 \otimes 1$ or $\Delta(1) = 1 \otimes ...
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50 views

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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1answer
53 views

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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1answer
83 views

General differentials operators (Grothendieck definition) and polynomial rings

Let $A$ be an algebra over some field $\mathbb{k}$. A linear map $f:A\to A$ is said to be a differential operator of an order $\le n$ if for all $a_0,a_1,\ldots a_n\in A$ we have ...
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61 views

on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, semi-simple); call $T$ its maximal torus, its Lie algebra $\operatorname{Lie}(G)=\mathbf g$, its Cartan subalgebra ...
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1answer
32 views

What is the natural action of $\mathfrak{sl}(4,\Bbb{C})$ on $\wedge^2 \Bbb{C}^4$?

What is the natural action of $\mathfrak{sl}(4,\Bbb{C})$ on $\wedge^2 \Bbb{C}^4$? We know that $\wedge^2 \Bbb{C}^4$ is generated by $\{e_1 \wedge e_2, e_1 \wedge e_3, e_1 \wedge e_4, e_2 \wedge e_3, ...
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1answer
144 views

Differential in Lie groups

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 ...
9
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1answer
173 views

Partial derivatives on Manifolds

Let $F : A \times B \to C$ be a map of smooth manifolds. Define the following maps ("partial derivatives"): $E_1 F: TA \times B \to TC$ $E_1 F(a,v,b) = D_a F(-,b) v $ where $v \in T_a A$ $E_2 F: A ...
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40 views

An alternative proof for the units of $U_q(\mathfrak{sl}_2)$ using Ore extensions.

I would like to establish what the set of units are in the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$. First, I recall the definition of the quantized enveloping algebra- throughout the ...
7
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2answers
302 views

Matrix exponential converse. Baker-Campbell-Hausdorff

I am currently reading about the Baker-Campbell-Hausdorff formula and in a textbook on Lie Algebras, he shows that if $$[X,[X,Y]] = 0 \quad \text{ and } [Y,[X,Y]] = 0$$ then $$e^{Xt}e^{Yt} = e^{Xt ...
2
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1answer
53 views

What is a simple lie algebra?

What is a simple lie algebra? What should I be thinking of when I come across these? What is a good example or two that I should keep in the back of my mind at all times? I know they are useful, but ...
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94 views

duality for (co)homology of Lie algebras

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. What is the relationship between $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, ...
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1answer
108 views

Lie algebras and the Killing form.

The Killing form is defined by $K(x,y) = \text{tr}(\text{ad} x, \text{ad} y)$, right? In this lecture, we assume that $\{x_1, ... , x_n\}$ is a basis for $g$ and $\{y_1, ... ,y_n\}$ is a dual basis ...
2
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1answer
62 views

Correspondence between one-parameter subgroups of G and TeG

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
2
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1answer
152 views

Quaternions as a Lie algebra, its derivations

Let $\mathbb{H}$ be the algebra of quaternions. It can be proven that each derivation $D:\mathbb{H}\to \mathbb{H}$ is inner that is of the form $\mathrm{ad}x$ for some $x\in \mathbb{H}$. I am to prove ...
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1answer
149 views

Two definitions of left-invariant vector fields of a Lie group

I am reading these lines from a text which shows why the bracket of two left-invariant vector fields is also a left-invariant vector field. But cannot easily get one of the lines. Let $L_a$ be the ...
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1answer
36 views

Matrices of subrepresentations and quotient representations.

Suppose that $V$ is a $5$ dimensional representation (with generators $\{y_1, ... , y_5\}$ of the lie group $\mathcal{g}$, with the lie algebra homomorphism $\rho: \mathcal{g} \rightarrow ...
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32 views

Computations of common isometry groups, $O(n)/O(n-1), SO(n)/SO(n-1), U(n)/U(n-1)$, etc?

On wikipedia, some of the common isometry groups are given: $S^{n-1}\cong O(n)/O(n-1)$, $S^{n-1}\cong SO(n)/SO(n-1)$, etc. Is there a reference where some/any of these groups are computed? I'm just ...
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1answer
223 views

What does the logarithm of the identity look like on a Lie group?

Let $G$ be a compact, connected Lie group with identity element $e$ and $\mathfrak g$ its Lie algebra. Consider the set $$ L=\{A\in\mathfrak g\setminus\{0\};\exp(A)=e\}. $$ The most descriptive name ...
2
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1answer
146 views

Different definitions of Casimir element

I read about the Casimir element just recently and I found it a bit difficult to wrap my mind around the definition(s). In fact, I have seen two different definitions. For concreteness, let ...
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51 views

Why do Ad(K) orbits in the $-1$ eigenspace of a Cartan decomposition intersect the Weyl chamber once?

Let $G$ be a semisimple Lie group and let $\frak p\oplus t$ be a Cartan decomposition of $\frak g$ and $K$ the connected subgroup with Lie algebra $\frak t$. Choose a maximal abelian subalgebra ...
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1answer
57 views

Reflection in terms of simple reflections

Suppose $\beta=\sum_{i=1}^ka_i\alpha_i$, where $\alpha_i$ are simple roots. Is there any easier way to write the reflection corresponding to $\beta$ say $s_{\beta}$ in terms of $s_{\alpha_i}$'s. I ...
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102 views

How to show $\exp(tX)\exp(tY)=\exp(t(X+Y)+tR(t))$ with $\displaystyle \lim_{t\to 0} R(t)=0$?

Let $X\in GL(n, \mathbb R)$. The exponential of $X$ is the matrix given by $$\exp(X)=\sum_{n=0}^\infty \frac{X^n}{n!}.$$ I need some help for showing the following result: ...
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81 views

Derivative of exponential map

Somehow I've gotten myself confused trying to take the derivative of the exponential map on $\mathfrak{so(3)}$. For vector $\theta$, $\delta \theta$, and $p \in \mathbb{R}^3$, define $$R(\theta, p) ...
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1answer
24 views

derivative of composition of rotations

Let $\theta$ and $\psi$ be two vectors in $\mathbb{R}^3$. I want to compute $$\nabla_{\psi} \log \left( e^{[\theta]_\times}e^{[\psi]_\times}\right)$$ Where $[v]_\times$ is the skew-symmetric ...
2
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1answer
82 views

Is there a natural Lie bracket for $\mathfrak X (M) \times C^\infty(M)$ (pairs of vector fields and smooth functions)?

Space of smooth vector fields $\mathfrak X(M)$ on a manifold $M$ has a structure of Lie algebra with the bracket being a commutator of two vector fields. Does cartesian product $\mathfrak X (M) ...