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.
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1answer
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Proving the Poisson differential is a coboundary map
Let $(M, \pi)$ be a Poisson manifold, that is, $M$ is a smooth real manifold and $\pi \in \mathfrak{X}^2(M)$ is a (possibly degenerate) skew-symmetric bivector field satisfying $[\pi, \pi] = 0$. Here ...
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2answers
19 views
What is the definition of polynilpotent Lie algebras?
I am looking for definition of polynilpotent Lie algebras. Is there any equivalent concept for that?
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0answers
24 views
Examples about “union of two Lie algebra is not Lie algebra” [on hold]
Could you give a some examples about "union of two Lie algebra is not Lie algebra".
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0answers
24 views
for arbitrary vector $v,u$, is there the matrix X which satisfy the relation exp$[X]\,v=u$?
Nowadays, I'm studying for exponential map of Lie group.
my question is, To make the form of exp$\begin{pmatrix}x_{11}&x_{12}&\cdots \\x_{21}&\ddots \\ \vdots\end{pmatrix}$,
I have to ...
2
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0answers
22 views
Is a stabilizer subgroup a symmetric subalgebra?
Consider the Lie algebra $\mathfrak{su}(n)$ and the set of operators that do not change the direction of the vector $\psi$,
$$
K:=\{ s\in \mathfrak{su}(n)\ :\ s \psi \propto \psi \}.
$$
Let $P$ be ...
0
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1answer
18 views
what is a left nilpotent Leibniz algebra
Let consider $L$ be a Leibniz algebra which is left nilpotent. ( I do not know what is left nilpotent of class 3).
A Leibniz algebra L is said to be nilpotent, if for lower central series there ...
1
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1answer
17 views
Show that $\mathfrak{b}_3 (\mathbb{C}) / \mathfrak{n}_3 (\mathbb{C}) \cong \mathfrak{t}_3 (\mathbb{C})$
Show that $\mathfrak{b}_3 (\mathbb{C}) / \mathfrak{n}_3 (\mathbb{C}) \cong \mathfrak{t}_3 (\mathbb{C})$, in which $\mathfrak{b}_3 (\mathbb{C}),\mathfrak{n}_3 (\mathbb{C}),\mathfrak{t}_3 (\mathbb{C})$ ...
0
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1answer
23 views
Doubt in the proof of Ado's theorem
I am currently going through a proof of Ado's theorem. I am stuck in one step.
Suppose $\mathfrak{g}$ is a solvable Lie algebra which is not nilpotent. Then one can show that there is an ideal $\...
4
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1answer
114 views
Getting representations of the Lie group out of representations of its Lie algebra
This is something that is usually done in QFT and that bothers me a lot because it seems to be done without much caution.
In QFT when classifying fields one looks for the irreducible representations ...
0
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1answer
14 views
Determine the derived series of $\mathfrak{b}_n (\mathbb{C})$.
Problem: Determine the derived series of $\mathfrak{b}_n (\mathbb{C})$, in which $\mathfrak{b}_n (\mathbb{C})$ is the space of all upper triangular matrices.
We knew that the derived series of a Lie ...
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0answers
12 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???...
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0answers
26 views
Why subalgebras containing engel algebras are self normalizing [duplicate]
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
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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 ...
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0answers
23 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 ...
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0answers
32 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 ...
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0answers
18 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 &...
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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
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1answer
34 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 ...
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0answers
39 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=\...
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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
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1answer
31 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. ...
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0answers
20 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 ...
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0answers
32 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 $\...
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2answers
41 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 ...
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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
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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
64 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
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1answer
21 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
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1answer
93 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 ...
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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
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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
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1answer
45 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 $G$ ...
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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 ...
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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!
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0answers
23 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....
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0answers
55 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 ...
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0answers
14 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
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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
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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 $W(E_6)\subset G$....
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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 $\...
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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 $n$....
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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 ...
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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
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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 ...
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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 ...
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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
$...
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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
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1answer
40 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 ...
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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
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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 ...
