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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Clarification of Definition: Free Algebra

I need some clarification on the definition of free algebra. Here is an extract from Lie Algebras and Lie Groups by Jean-Pierre Serre: I am somewhat confused about the definition of free algebra. ...
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80 views

Reference request: classification of simple Lie groups and simple real Lie algebras

I am trying to understand the classification of simple Lie groups and the theory of highest weights for semisimple Lie groups by first understanding the case for complex Lie algebras, then relating to ...
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174 views

Convolution of matrix coefficients is also a matrix coefficients

I have a question about the convolution of matrix coefficients as follows: Let $G$ be a compact Lie group. A Map $f:G\rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite ...
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40 views

How can the existence of this expression with Cartan matrix be shown using Killing form?

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra. Let $\mathfrak{h}$ be a Cartan subalgebra, $C$ the Cartan matrix, and $R$ a system of simple roots ...
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What do you get when you pull the Bruhat Decomposition back to the Lie algebra via the exponential map?

If $G$ is a connected, reductive, complex group with Borel subgroup $B < G$ and Weyl group $W$, we can write $$G = \bigsqcup_{w \in W} B w B$$ If $\mathfrak{g}$ is the Lie algebra of $G$, we have ...
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47 views

Isomorphic Lie algebras have isomorphic centers

I think that if two Lie algebras are isomorphic, then their centers should be isomorphic - is this true? I am sure the answer is obvious to those in the know! Here is my attempt at a proof which looks ...
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25 views

Is $\mathfrak{o}(n)$ a subalgebra of $\mathfrak{u}(n)$?

A quick simple question to start the weekend (I hope). The Lie algebra $\mathfrak{u}(n)$ is the set of $n\times n$ skew-Hermitian matrices over $\mathbb{C}$ and the Lie algebra $\mathfrak{o}(n)$ is ...
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61 views

How to determine the number of roots and the dimension of a Lie algebra using Cartan matrix

Let $\mathfrak{g}$ be a Lie algebra with the Cartan matrix $$ C=\left(\begin{array}{cc} 2 & -2\\ -1 & 2 \end{array}\right) $$ Question: How can the number of roots of $\mathfrak{g}$ be ...
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29 views

Finding free subgroups thanks to Lie algebras

Let $f : F \to G$ be a homomorphism from a free group $F$ to a group $G$. I heard that, in order to verify whether or not $f$ is one-to-one, it is possible to associate a Lie algebra $E_0^*(H)$ to any ...
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81 views

Lie Bracket of vector fields on Lie group

Let $H$ be a Lie group and $\mathfrak{h}$ its Lie algebra. Given a smooth function $v: H \to \mathfrak{h}$, define the vector field $\bar{v} : H \to TH$, $h \mapsto d(R_{h})_{e} v(h)$, where $R_{h} : ...
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93 views

Relation between Aut(G) and Aut(g)

Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. We know that when $G$ is simply connected, $\mathrm{Aut}(G)=\mathrm{Aut}(\mathfrak{g})$ (this should follow from the fact that we can ...
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70 views

Examples of Free Lie Algebra

In wikipedia, free Lie algebras are defined using the universal property. Can anyone give some concrete examples of free Lie algebras? "In mathematics, a free Lie algebra, over a given field $K$, is ...
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84 views

Does the matrix exponential take open sets into open sets?

This is from Hall's Lie Groups, Lie Algebras, and Representations, in theorem $2.13$: Let $B_\varepsilon$ be the open ball of radius $\varepsilon$ about zero in $M_n (\mathbb{C})$ [$= ...
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46 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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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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27 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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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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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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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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83 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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65 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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151 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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118 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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267 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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73 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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41 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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127 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 ...
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89 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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146 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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60 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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445 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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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 ...
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284 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 ...
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412 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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49 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 ...
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1answer
106 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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34 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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69 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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59 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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59 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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95 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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62 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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33 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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156 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 ...
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180 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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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 ...
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253 views

$\operatorname{span}\{AB-BA : A, B \in M_n(\Bbb R)\}$ is the set of all matrices with trace $0$

Let $W$ be the space $$\operatorname{span}\{AB-BA\} ,$$ where A and B are square matrices, and let $H$ be the space of all square matrices of trace $0$. Then prove that $W=H$. The fact that $W$ ...
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342 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 ...
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1answer
57 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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95 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 $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, ...