Tagged Questions

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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2
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1answer
29 views

Determining Lie algebras from commutative diagrams of exact sequences

Suppose that we have the following commutative diagram of graded Lie algebras $$\begin{array} A 0& {\longrightarrow} & C_n & {\longrightarrow} & A_{n+1} &{\longrightarrow} & ...
1
vote
1answer
9 views

The Weyl group and eigenspaces

Let $V$ be a representation of the Weyl group. For any reflection $\sigma_{\alpha}$ (where $\alpha$ is a root), we know that $V$ has two eigenspaces with eigenvalues $1$ and $-1$. The ...
1
vote
1answer
22 views

polynomial algebras and their coefficients in prime fields

According to the definition of the polynomial algebras $A(n)$ and $A(n,m)$ for $ n \in \mathbb {N} $ and $ m \in \mathbb {N}^n$, if $\mathbb{F}$ be field $GF(2)$ and $ X_1,...,X_n$ be $n$ pairwise ...
1
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0answers
15 views

Normal subgroup invariant under $\text{Ad}_g$

Denote by $G$ a Lie group with corresponding Lie algebra $\text{Lie}(G)$. There the three maps inner automorphism/conjugation: $\text{Int}_g = L_{g^{-1}} \circ R_g \in \text{Aut}(G)$, $\text{Ad}_g ...
2
votes
2answers
35 views

The universal enveloping algebra of free Lie algebra is the tensor algebra on the Free Abelian Group?

Let $A$ be a set of cardinality at least 2 and let $M_A$ be the free abelian group generated by $A$. Let $L(A)$ be the free Lie algebra generated by $A$. I am reading On Injective Homomorphisms For ...
3
votes
1answer
29 views

The action of a Lie algebra on a manifold is a Lie algebra homomorphism. How to show it?

By definition, the action of a Lie algebra $\mathfrak g$ on a manifold $M$ is a Lie algebra homomorphism, $\mathcal A: \mathfrak g\rightarrow\mathfrak X(M), \xi\mapsto\xi_M$ such that the action map ...
1
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0answers
9 views

$H^q(\mathfrak{g},K;V)$ is equal to $Ext_{\left(\mathfrak{g},K\right)}^q\left(\mathbb{C},V\right)$?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $K$ be a closed subgroup of $G$ with corresponding Lie subalgebra $\mathfrak{k}$. Let $V$ be a $\left(\mathfrak{g},K\right)$-module. Then, I ...
0
votes
2answers
65 views

What is the nilradical of $\mathfrak{gl}_n$?

I'm really embarrassed to ask but what is the nilradical of the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$, i.e. the set of ad-nilpotent elements of $\mathfrak{gl}_n(\mathbb{C}) = ...
1
vote
1answer
24 views

Is the center of a compact Lie algebra precisely the set of vectors on which the Killing form is zero?

Suppose a Lie algebra $\frak{g}$ has a killing form, $B$, which is negative semidefinite. Suppose $B(X,X)=0$ for some $X\in \frak{g}$. Is $X$ necessarily in the center of $\frak{g}$?
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0answers
24 views

What is a Serre presentation of a Lie algebra?

For example, as in: Give a Serre presentation of Lie algebra $\frak{g}$ of type $G_{2}$. Is it the presentation in terms of Chevalley generators, which satisfy Serre relations?
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0answers
26 views

Partial generalisation to Whitehead's second Lemma

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional semisimple $k$-Lie algebra. By Whitehead's second Lemma, we know that $H^{2}(\mathfrak{g}, ...
1
vote
1answer
22 views

How does one define weights for a semisimple Lie group?

For compact Lie groups one considers a maximal torus to define the weight space decomposition of a representation. For a complex semisimple Lie algebra one considers a Cartan subalgebra. How does ...
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0answers
17 views

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. ...
1
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0answers
18 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 ...
4
votes
0answers
37 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 ...
1
vote
1answer
11 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 ...
0
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0answers
10 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 ...
1
vote
1answer
30 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 ...
1
vote
1answer
18 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 ...
0
votes
1answer
9 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 ...
1
vote
1answer
19 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 ...
1
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0answers
38 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} : ...
3
votes
1answer
66 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 ...
2
votes
1answer
36 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 ...
0
votes
1answer
48 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 $M_n (\mathbb{C})$ [$= ...
2
votes
1answer
15 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 ...
0
votes
0answers
13 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 ...
1
vote
1answer
16 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} ...
0
votes
0answers
22 views

$[L,L]$ nilpotent means L solvable

How can I prove that if a lie algebra $L$ satisfies that $[L,L]$ is nilpotent, it is solvable? I know that the converse is one of Lie's theorem corollaries.
4
votes
0answers
37 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 ...
1
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0answers
31 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 ...
2
votes
0answers
44 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 ...
0
votes
0answers
28 views

Root Systems and Dynkin diagrams.

On page 142, the textbook An Introduction to Lie Groups and Lie Algebras (by Kirillov) determines the fundamental group of the root system $A_2$. Basically, the author says we have two simple roots ...
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0answers
39 views
2
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0answers
20 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 ...
1
vote
1answer
48 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 ...
5
votes
1answer
20 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 ...
0
votes
0answers
10 views

SO(2,1) invariance algebra.

Please excuse me for my ignorance. I would like to know how $SO(2,1)$ Lie algebra is derived from operators and commutators. I have some notes, that the Lie algebra of $SO(2,1)$ is derived from: ...
0
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0answers
11 views

Gauge theory H(P) and V(P)

In general you get a connection out of a one-form $\omega$ where $\omega = g^{-1}dg+g^{-1}Ag$ and $A= A^{\alpha}_{\mu}\frac{\lambda_{\alpha}}{2i}dx^{\mu}$ is given and the base ...
0
votes
0answers
25 views

Split exact sequence of Lie algebras

Suppose that we have the following commutative diagram of Lie algebras $$\begin{array} A 0& {\longrightarrow} & A_0 & {\longrightarrow} & A_1 &{\longrightarrow} & A_2 & ...
0
votes
0answers
12 views

How to show that $U(\mathfrak{n})^* \cong \mathbb{C}[U]$?

Let $U$ be the unipotent subgroup of $GL_n(\mathbb{C})$ consisting of all unipotent upper triangular matrices. Let $\mathfrak{n}$ be its Lie algebra. I heard many times that $U(\mathfrak{n})^* \cong ...
0
votes
0answers
47 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 ...
1
vote
2answers
48 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 ...
1
vote
1answer
30 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 ...
0
votes
1answer
33 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
votes
1answer
31 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 $\operatorname{ad}$, ...
1
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0answers
40 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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0answers
34 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 ...
1
vote
2answers
65 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 ...
1
vote
1answer
23 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 ...