Questions tagged [lie-algebras]
For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.
3,974 questions
0
votes
0answers
9 views
Show that $L/L'$ abelian
Problem: Let $L$ be a Lie algebra, denote $[L L]=L'$. Show that $L/L'$ abelian.
My attempt: $[x,y] = (x+L')(y+L')-(y+L')(x+L') = ((x+y)+L') - (y+x+L') = ((x+y)+L') - ((x+y)+L') = 0$
Is that enough???...
0
votes
0answers
14 views
Why subalgebras containing engel algebras are self normalizing
It is a fact in Humphrey's book that subalgebras containing Engel algebras are self normalizing.
This was already asked here:
Help with proof in Humphreys (2)
But the accepted answer doesn't ...
1
vote
1answer
20 views
Lie algebras of infinite dimensional Lie groups
I have to work with Lie algebras of some infinite dimensional 'Lie groups' (e.g. $\Omega SL_2(\mathbb{C})$) but i'm not sure on how to approach infinite dimensional groups, for loop group it is not so ...
1
vote
0answers
16 views
References for Lie algebra extensions, Poincaré
I will be posting this to the physics stack since this is a physics paper, but I figured the mathematicians would be able to suggest more comprehensive references for me.
My future adviser just ...
0
votes
0answers
30 views
What is the notation $\delta$ here?
I'm reading the book "Introduction to Lie Algebra and Representation Theory - J. E. Humphrey", I have a question on an example on the page number $2$. That is
Example: For reference, we write down ...
1
vote
0answers
16 views
$SL_2 (\Bbb R) × SL_2 (\Bbb R)/ ± (I_2 , I_2 ) → (SO_{2,2})^\circ$
I went through this problem in Lie groups:
i) Prove that $SL_2 (\Bbb R) × SL_2 (\Bbb R)$ is a linear Lie group.
I identified $SL_2 (\Bbb R) × SL_2 (\Bbb R)$ with $\{\begin{pmatrix}
A & 0 \\
0 &...
0
votes
1answer
13 views
$\operatorname{coker}(\phi)$ is discrete for a morphism of Lie groups
suppose $\phi: G \to H$ a morphism of Lie groups such that $d\phi$ is surjective.
Prove that $\operatorname{coker}\phi$ is discrete.
My attempts:
Prove that $\phi(G)$ is open which will lead to $H/\...
1
vote
1answer
31 views
A basic example to understand the concept of “Weight”
Let $A=b(2,\mathbb{R})$ be he Lie subalgebra of upper triangle matrices of $gl(2,\mathbb{R})$. It is clear that $e_1=(1,0)$ is an eigenvector for $A$, because it is an eigenvector for every element of ...
1
vote
0answers
35 views
What are the possible Lie algebras of $K=\rho(\mathbb{R}^2)$, where $\rho :\mathbb{R}^2\to \text{Aff}(\mathbb{R}^2)$?
I am reading a paper of Yves Benoist (Tores Affines) and I can't figure out how to answer the following question.
Let $\rho :L\to \text{Aff}(\mathbb{R}^2)$ be a Lie group homomorphism, where $L=\...
0
votes
0answers
38 views
Relation between Symmetric algebra and Universal enveloping algebra as Lie algebras.
Let $L$ be a Lie algebra over $\mathbb{C}$. Assume $L$ satisfies PBW theorem. We can associate two Lie algebras with $L$:
1) $U(L):$ the universal enveloping algebra. Here the Lie bracket is defined ...
1
vote
0answers
24 views
When is exponential map from Lie algebra to Lie group a covering map?
Suppose $G$ is a Lie group and $\mathfrak{g}$ its Lie algebra. It is not so difficult to see that if $G$ is abelian and connected then $\exp:\mathfrak{g}\rightarrow G$ is a universal covering map. ...
0
votes
0answers
19 views
relation between trace and hat operator (skew-symmetric matrices)
To avoid confusion, let me first introduce the notation (although pretty standard) which is required for the question that I want to ask. Let $\mathsf{GL}(3,\mathbb{R})$ be the set of $3\times 3$ real ...
0
votes
0answers
31 views
Definition of action of Lie algebra of an algbraic group
Here is the context of my interrogation :
Let $G$ be an affine algebraic group over $\mathbb{C}$ acting rationally on an affine variety $X$ over $\mathbb{C}$.
This induces an action of $G$ on $\...
1
vote
2answers
39 views
Generating set of lie algebra su(3)
I am looking for (an example of) a minimal set of Gell-Mann matrices such that their closure under the Lie bracket is all of $\mathfrak{su}(3)$. By minimal I mean the set should be as small as ...
0
votes
0answers
18 views
commutator ideal for direct sum
Is commutator ideal compatible with direct sum?
Let's take $\mathfrak{sl}_2(\Bbb K)\oplus\mathfrak{sl}_2(\Bbb K)$ which is semi-simple because $\mathfrak{sl}_2(\Bbb K)$ is simple Lie algebra.
So we ...
0
votes
1answer
20 views
Adjoint representation of $\mathfrak{b}_2$ is undecomposable
Let $\mathfrak{g}=\mathfrak{b}_2(\Bbb C)$.
Prove that adjoint representation $ad_\mathfrak{g}$ of $\mathfrak{g}$ is undecomposable into a direct sum of irreducible representations.
My attempt:
I ...
2
votes
2answers
63 views
Question about linear algebraic groups split vs isotropic
I am reading notes on linear algebraic groups and I'm getting confused with some definitions and I would appreciate any clarification.
They define $G$ to be split if there exists a maximal torus $T$ ...
1
vote
1answer
20 views
adjoint representation is irreducible iff $\mathfrak{g}$ is simple
I am trying to prove that for a Lie algebra $\mathfrak{g}$:
$ad_{\mathfrak{g}}$ the adjoint representation of $\mathfrak{g}$ is irreducible iff $\mathfrak{g}$ is simple.
I tried to use the fact that ...
5
votes
1answer
92 views
Is $SL(n,\mathbb{R})/SL(n, \mathbb{Z})$ a Hausdorff space?
The special linear group $SL(n, \mathbb{R})$ of degree $n$ over $\mathbb{R}$ is the set of $n \times n$ matrices with determinant $1$, with the group operations of ordinary matrix multiplication and ...
1
vote
0answers
32 views
Simple Lie algebra representations and tensor powers of fundamental representations [duplicate]
Let $\frak{g}$ be a simple Lie algebra over $\mathbb{C}$. We will call a representation of $\frak{g}$ tautological if it is a fundamental representation of smallest dimension.
For $V$ a tautological ...
2
votes
0answers
68 views
If $X,Y$ generates $\mathfrak{g}$ then $e^{tX}$ and $e^{tY}$ generates de Lie Group G.
I'm trying to solve the following problem of the book "Grupos de Lie - Luiz A. B. San Martin":
Question: Let $G$ be a connected Lie Group with Lie algebra $\mathfrak{g}$. Suppose that $X,Y \in \...
3
votes
1answer
42 views
Lie group-algebra representations
I want to prove the following: Given two representations of a connected matrix Lie group are equivalent if and only if the associated Lie algebra representations are equivalent.
Definition: Let $G$ ...
0
votes
0answers
17 views
The representation $\Phi_{n} $ of $SU_{2}$ is irreducible.
The proof is given below:
But I do not understand why " we first determine which of the spaces of $V_{n}$ are invariant under T " as he said in the second sentence. could anyone explain this for ...
0
votes
0answers
5 views
A discrepancy in a paragraph in the construction of a series of irreducible complex representations of $SU_{2}$
The construction is given below:
But I do not understand the last sentence and the paragraph before it, Could anyone give me a concrete example to explain them please?
Thank you!
1
vote
0answers
20 views
Relation between semisimple Lie algebra completely reducibility and semisimple ring
Let $\frak g$ be a semi-simple finite dimensional Lie algebra over the complex numbers $\mathbb C$. Then every non irreducible representation of $\frak g$ is completely reducible.
Q1: Is category f....
2
votes
0answers
54 views
Two commutator relations.
$\newcommand{\ad}{\operatorname{ad}}$Let $R$ be an associative ring. Set $[x, y] = xy - yx$ and $\ad_x(-) = [x, -]: R \to R$.
Is there a formula for $(ab)^n$ in general? I found one formula for the ...
0
votes
0answers
13 views
How is the Lie bracket defined for two elements in the dual (cotangent) space?
If $\xi\in se(3)$ and $\xi^{*}\in se^{*}(3)$. How can we define $[\xi^{*}_{1},\xi^{*}_{2}]$?
4
votes
1answer
33 views
motivation of coalgebra
I don't know why should we need coalgebra?What is the motivation?By changing all the arrows of algebra structure, it seems strange.What is the application of coalgebra? What is the relation with Lie ...
1
vote
1answer
24 views
The automorphism group of the lattice $E_6$
Let $E_6$ be the root lattice, and $G$ be its automorphism group as a lattice (i.e. as $\mathbb Z$-module together with the inner product). Let $W(E_6)$ be the Weyl group. Apparently $W(E_6)\subset G$....
2
votes
0answers
14 views
Reducing the Dimensionality of the Sphere in terms of the Lie Algebra
The $n$-sphere can be written as an $(n-1)$-sphere fibered over an interval
$$
ds^2_{\Omega_n} = d\theta^2 + \sin^2 \theta\;d\Omega_{n-1}^2.
$$
In these coordinates, when we impose that we keep $\...
0
votes
0answers
11 views
Are these enough conditions for the subgroup to be a full latice?
I was wondering how to prove the following, or if you like, whether it is true, although I am almost certain it is. Let $V$ be a finite dimensional vector space over $\mathbb{C}$, say of dimension $n$....
0
votes
1answer
21 views
Verifying that the given map defines a Lie algebra
I am given a matrix $A =(a,b;c,d)$ in $GL(2,\mathbb{C})$ and a real algebra say, $V$ with basis $X,Y,Z$ such that $[X,Y]=0, [X,Z]=aX+bY, [Y,Z]=cX+dY$ I have to show that $V$ is a real Lie Algebra.
My ...
0
votes
1answer
9 views
$Ker(exp_G )$ is discrete
$G ⊂ GL_n (\Bbb R)$ is an abelian connected Lie group, $\mathfrak{g}$ its Lie algebra and $exp_G : \mathfrak{g} → G$ the exponential map.
Prove that $Ker(exp_G )$ is discrete.
My attempt:
$Lie(Ker(...
2
votes
1answer
25 views
dφ is bijective but φ is not a lie group isomorphism
suppose G and H connected Lie groups. Is there $\phi: G \to H$ a morphism of Lie groups such that $d\phi$ is bijective but $\phi$ is not an isomorphism of Lie groups?
I know that $d\phi$ surjective ...
1
vote
0answers
10 views
Expressing every element of a lie group as a product of exponentials?
Given a connected lie group $G$, since a neighbourhood of the origin generates all of $G$, we have that every $g$ in $G$ can be expressed as a finite product of elements of the form $e^X$, for $X$ in ...
0
votes
0answers
21 views
induced morphism between lie groups is surjective
Let $φ : G → H$ be a continous morphism of linear Lie groups with H connected.
Prove that $φ$ is surjective iff $dφ$ surjective
Using the expression : $exp\circ dφ = φ \circ exp$ we can see that
$...
0
votes
0answers
16 views
Skew symmetric matrix of vector follow up - How to obtain skew matrix for N-dimensional vector
I am trying to understand usage of skew matrices:
1) It is my understanding from an earlier question in this forum that for vectors A and B, that A cross B is the same as the "skew symmetric matrix" ...
0
votes
1answer
39 views
Dimension of image of one parameter subgroup. [closed]
If $G$ is a Lie group then $\eta : \mathbb{R}\to G$ is called one parameter subgroup if it is a continuous group homomorphism.
I need to show that the images of one-parameter subgroups in a Lie group ...
-4
votes
1answer
82 views
Show that every irreducible representation of $SO_{3}$ is isomorphic to one of the representations $\Psi_{n}$.
The question is given below:
And this is the mentioned exercise:
And this is 7.4:
Could anyone give me a hint about the solution of the question, I am stucked in it ?
2
votes
0answers
40 views
The automorphism group of a Lie algebra
Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra. It is well-known that the the automorphism group of $\mathfrak{g}$, $\operatorname{Aut}(\mathfrak{g})$, is an algebraic group.
How ...
0
votes
0answers
11 views
If $G$ is a Lie group and $H$ is a closed Lie subgroup, then $G\to G/H$ is a principal- $H$ bundle.
Let $G$ be a Lie group and $H$ be a closed Lie subgroup of $G$. Let $G/H$ has the quotient topology. Then $ p: G\to G/H$ is a principal-$H$ bundle.
I was reading the above theorem from the book ...
2
votes
1answer
49 views
Use the theory of characters to derive the following relation for the representations of $SU_{2}.$
The question is given below:
And the hint at the back of the book says:
Establish the corresponding equality for characters.
And this was a question I was helped on it, which establish the relation ...
-1
votes
1answer
17 views
Derivative of a curve in a Lie group
This is probably a trivial question, but I am not able to see how to go about it:
Let $G$ is a Lie group and let $\gamma : I \subset \mathbb{R} \rightarrow G$ is a smooth curve in $G$. Is it true ...
3
votes
2answers
160 views
What is an example of an abelian Lie group $G$ and a closed subgroup $H$ such that $G\not\cong G/H \times H$?
What is an example of an abelian Lie group $G$ and a closed subgroup $H$ such that $G\not\cong G/H \times H$?
Would the circle $S^1$ in $\mathbb R^2$ be an example? what is $\mathbb R^2/S^1$?
4
votes
1answer
57 views
Inner automorphisms of a real semisimple Lie algebra
There are at least two ways of defining the inner automorphisms of a real Lie algebra $\mathfrak{g}$. One is the algebraic definition: an inner automorphism is $\exp (\text{ad} X)$, where $X$ is an ...
0
votes
0answers
77 views
A discrepancy in understanding a solution given to me here for a problem of Vinberg section 8.
Prove that the linear span of the functions: $$\phi_{n}(z) = \chi_{n}(A(z)),(z \in \mathbb{C}, |z| = 1) $$
coincide with the space of all functions $\phi$ on the unit circle which can be written as ...
1
vote
0answers
24 views
Lattice and abelian Lie groups
Let $\Lambda$ be a discrete (lattice) subgroup of $\mathbb R^n$. Let $V:=\langle \Lambda\rangle_\mathbb R$. Define the abelian Lie group $G:=V/\Lambda$.
Now if $H$ is a Lie subgroup of $G$. Does ...
1
vote
0answers
92 views
Show that the characters of the representations $\phi_{n}$ of $SU(2)$ constitute a complete orthogonal set.
The question is given below:
And the other questions mentioned are (I know the solutions of all of them):
Sorry for the bad formulation of the my question at the first time I have ...
8
votes
0answers
216 views
+50
Complexification of compact connected Lie groups: do these curves have the same tangent vector?
I'm trying to understand the complexification of Lie groups from page $207$ here and I'm having trouble with a computation.
Assume $A, B$ are hermitian metrices, and $k$ is a unitary matrix. I want ...
2
votes
0answers
26 views
Computing matrix logarithm and exponential.
I am solving exercise of 5.11.3 in Lie groups, Lie algebras, and Representations by Brain Hall.
The logarithm and exponential are defined by
\begin{equation}
\log X = \sum_{m=1}^{\infty} (-1)^{m+1} \...
