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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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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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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 ...
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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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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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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 ...
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$SL_2 (\Bbb R) × SL_2 (\Bbb R)/ ± (I_2 , I_2 ) → (SO_{2,2})^\circ$

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 $\{\begin{pmatrix} A & 0 \\ 0 &...
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1answer
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$\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/\...
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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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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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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 ...
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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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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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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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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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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 ...
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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 ...
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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 $T$ ...
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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 ...
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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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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 ...
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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 \...
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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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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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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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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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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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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 $[\xi^{*}_{1},\xi^{*}_{2}]$?
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1answer
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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 ...
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1answer
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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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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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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
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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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$Ker(exp_G )$ 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(...
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1answer
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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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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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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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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" ...
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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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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 ?
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\mathbb R^2/S^1$?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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} \...