Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
1answer
111 views

Uniqueness of the Lie brackets in the quotient space of a Lie algebra

Suppose I have a Lie algebra $\mathfrak g$ which has an ideal $\mathfrak a$. Then I consider the quotient set $\mathfrak g / \mathfrak a$ which is the set of all equivalence relations of $\mathfrak g$ ...
0
votes
1answer
8 views

Quotient algebras of nilpotent Lie algebra are nilpotent

For the following proposition I found a proof in some notes that I don't understand. Below definition 1 defines the terminology I'm using, and proof attempt 1 gives my attempt at the proposition. ...
0
votes
2answers
26 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?
1
vote
1answer
24 views

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 ...
0
votes
1answer
2k views

Basis of $SO(4)$ group

I have a rotation matrix $$ R_{(\phi)} = \left( \begin{matrix} \cos (\phi) & \sin (\phi) & 0 & 0 \\ -\sin(\phi) & \cos(\phi) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & ...
-2
votes
0answers
27 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".
1
vote
0answers
25 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
votes
0answers
23 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 ...
1
vote
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. ...
0
votes
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
vote
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 $\mathfrak{b}_3 (\mathbb{C}),\mathfrak{n}_3 (\mathbb{C}),\mathfrak{t}_3 (\mathbb{C})$ ...
4
votes
1answer
115 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
votes
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 $\...
0
votes
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 ...
0
votes
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 ...
0
votes
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???...
6
votes
1answer
1k views

Center of $\mathfrak{sl}(n,F)$

Prove that $\mathfrak{sl}(n,F)$ (matrices with trace zero) has center $0$, unless $\operatorname{char}F$ divides $n$, in which case the center is $\mathfrak{s}(n,F)$ (scalar multiples of the identity)....
0
votes
0answers
44 views

What is known about such dominant integral weights of compact semisimple Lie groups?

I am interested in special dominant integral weights $\lambda \in \mathfrak{h}^*$, where $\mathfrak{h}$ is a Cartan subalgebra of the Lie algebra $\mathfrak{g}$ of a compact semisimple Lie group $G$. ...
0
votes
0answers
42 views

Proving that $G=\mathrm{SL}_d(\mathbb{R})$ is semisimple

I want to find a reference to a proof of semisimplicity of the special linear group $G=\mathrm{SL}_d(\mathbb{R})$ by showing that the Lie algebra $\mathfrak{g}$ of $G$ of matrices with trace $0$ is ...
4
votes
1answer
63 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 ...
1
vote
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....
0
votes
0answers
23 views

Is this subalgebra semisimple?

Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Let $\Phi$ be the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by ...
1
vote
1answer
21 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
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 ...
0
votes
0answers
34 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 ...
10
votes
3answers
2k views

Reference for Lie-algebra valued differential forms

I am learning about vector-valued differential forms, including forms taking values in a Lie algebra. On Wikipedia there is some explanation about these Lie algebra-valued forms, including the ...
9
votes
4answers
1k views

Computing the Lie bracket on the Lie group $GL(n, \mathbb{R})$

Consider the Lie group $GL(n, \mathbb{R})$. Since $GL(n, \mathbb{R})$ is an open subset of the space $M_{n,n}(\mathbb{R})$ of $n \times n$ matrices, we can identify the tangent space (Lie algebra) $T_{...
1
vote
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 &...
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/\...
3
votes
2answers
77 views

Classification of real semisimple lie algebras

We know that each complex semisimple lie algebra $L$ is a direct sum of a chosen Cartan subalgebra $H$ and finitely many weight spaces, each of which is associated with an element in $H^*=\...
6
votes
0answers
268 views

Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
1
vote
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 ...
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
41 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
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 ...
8
votes
0answers
245 views

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
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 $T$ ...
-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 ?
5
votes
1answer
94 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
2answers
42 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
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 $\...
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 ...
1
vote
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 ...
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
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
69 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 \...
1
vote
1answer
51 views

What is the rigorous way for a Lie group (SU(n)) element to be “near” another element?

Statement of the problem I'm working with a function $\lambda : SU(n)\times SU(n)\times SU(n) \rightarrow \mathbb{C}$. Given $U_1, U_2, U_3 \in SU(n)$, I'd like to know how to calculate $\lambda (\...
3
votes
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 $G$ ...
2
votes
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 ...