For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

learn more… | top users | synonyms (1)

2
votes
1answer
143 views

establishing an isomorphism

I request help with this is a question from Introduction to Lie algebra by Erdmann and Wildon. The question asks to show that show that $so(4,\mathbf{C})\cong sl(2,\mathbf{C}) \oplus ...
4
votes
2answers
538 views

Is every skew-adjoint matrix a commutator of two self-adjoint matrices

I'm looking to solve some matrix equations. One of the equations involves a commutator, so my question is as follows: let $A$ be a skew-self-adjoint, traceless matrix, does the equation $[X,Y] = A$ ...
1
vote
1answer
65 views

Indecomposable L-module

I have the following exercice which I have be trying to solve: Let L be a Lie algebra and $r:L\rightarrow gl_3(F)$ a representation of L such that $im(r)=t_3(F)$ (the upper triangular matrices). Show ...
0
votes
1answer
46 views

Lie product of a two subalgebras

Let V and W be subalgebras of a Lie algebra $\mathcal{L}$. I want to show that $[V,W]$ is not always a subalgebra of $\mathcal{L}$.
5
votes
1answer
273 views

The Universal enveloping algebra of a finite dimensional Lie algebra is Noetherian.

If $\mathfrak{g}$ is a finite-dimensional Lie algebra, then it is very known that the Universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ is a Noetherian ring. What is the simplest way to ...
1
vote
0answers
113 views

Submanifold of a Lie group - tangent space

Let $G$ be a compact Lie group and $H, H' \leq G$ Lie subgroups. Consider the set $M = H' \cdot H = \{h\cdot h' \ \vert \ h \in H, h' \in H'\}$. Is it possible to describe explicitly the tangent space ...
4
votes
2answers
219 views

Does the proof of the Poincare-Birkhoff-Witt theorem need the Jacobi identity?

The title says it. Suppose I have a vector space $V$ equipped with a bilinear bracket such that $[x,y]=-[y,x]$, and define the universal enveloping algebra $U$ as usual: namely the tensor algebra on ...
2
votes
0answers
76 views

Dimension of Abelian Lie Algebras

I've tried to answer this question, but I need some help. What is the possible dimension of irreducible representations of Abelian Lie Algebras? I think it is always one, but I am not sure. Thank ...
0
votes
1answer
96 views

A question about Lie algebras corresponding to Lie groups and algebraic groups

Lie groups and algebraic groups both correspond with Lie algebras, which are by definition the left invariant vector field. But the topology of Lie groups and algebraic groups are different. Are their ...
3
votes
1answer
103 views

Sanity check on spider web calculation

For fun I began reading an interesting online paper I found, Spiders for rank 2 Lie algebras, and on page $5$ we have the following calculation, akin to a tensor product expansion via bilinearity: ...
4
votes
0answers
190 views

When is the adjoint representation self-dual?

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...
4
votes
2answers
197 views

Lie algebra 3 Dimensional with 2 Dimensional derived lie algebra

i read in mark wildon book , an introduction to lie algebras, in page 22 say that : Suppose that dim $L$ = 3 and dim $L'$ = 2. We shall see that, over $\mathbb{C}$ at least, there are infinitely many ...
0
votes
1answer
56 views

preserves eigen spaces?

"Let $H_0=\begin{pmatrix}i&0\\0&-i\end{pmatrix}$, suppose $A\in SU(2)$ commutes with $H_0$, it must preserves each eigen spaces for $H_0$, eigen spaces for $H_0$ are just $\mathbb{C}e_1$ and ...
2
votes
1answer
117 views

Weyl group of $\mathfrak{sl}(2,\mathbb{C})$

$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{gl}(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the ...
1
vote
0answers
73 views

Is there any Lie algebra that is not constructed from an associative algebra

I see in Wikipeida that every Lie algebra is either constructed from an associative algebra by defining: $[x,y]=xy-yx$, or a subalgebra of a Lie algebra thus constructed. Where can I find a proof? ...
1
vote
1answer
58 views

Different orderings for highest weights of a representation

Recall that given a representation $\pi$ of $\mathfrak{sl}_n$, a weight $\mu$ is said to be of highest weight if its corresponding weight vector is annihilated by all the positive root spaces (1). ...
-2
votes
1answer
101 views

Three dimensional Lie algebra L with dim L' = 1

Now suppose the derived algebra has dimension 1. Then there exits some non-zero $X_1 \in g$ such that $L' = span\{X_1\}$. Extend this to a basis $\{X_1;X_2;X_3\}$ for g. Then there exist scalars ...
0
votes
1answer
139 views

Cartan subalgebra of simple Lie algebra

I could not get the following, could someone give me a hint? Let $\mathfrak{H}$ be a Cartan subalgebra of a simple Lie algebra $\mathfrak{L}$. Show that $\mathfrak{H}$ is abelian. So, we need to ...
3
votes
0answers
105 views

finding highest weight of dual of a representation of a semisimple lie algebra

If $V$ is an irreducible representation of a semi simple lie algebra having highest weight $\lambda$ then what will be the highest weight of the corresponding irreducible representation $V^*$ (Dual of ...
1
vote
0answers
62 views

Followup question in Brian Hall's Lie Groups and Algebras.

In ex 9, page 60, he writes down that in order to prove that each invertible matrix $A$ can be written as $A=e^X$, where $X\in M_{n\times n}$, one need to use the fact that if $A$ is unipotent then ...
4
votes
2answers
662 views

Example of two-dimensional non-abelian Lie algebra?

can some one give me an example of two-dimensional non-abelian Lie algebra?
1
vote
2answers
885 views

Two Dimensional Lie Algebra

I read in mark wildon book "introduction to lie algebras" "Let F be any field. Up to isomorphism there is a unique two-dimensional nonabelian Lie algebra over F. This Lie algebra has a basis {x, y} ...
1
vote
0answers
54 views

Basics of Lie 2-algebras?

Could somebody (simply) explain the basics foundations of Lie 2-algebras, and some basic practical applications ? For instance, does it exist a 3-map (equivalent to the 2-map commutator for Lie ...
3
votes
1answer
269 views

showing that a lie algebra is the direct sum of two ideals.

I am trying to do an exercise(2.13) in Wildon and Erdmann's Intro to Lie Algebras and I'm stuck. The meat of the question is to show that under the following conditions i. the center, $Z(I)=0$ ii. ...
1
vote
1answer
110 views

Eigenvalues of a Lie bracket

Let $V$ be a complex vector space. Suppose that $a,b \in gl(V )$ (the set of all linear maps from $V$ to $V$) satisfies $$[a,[a,b]] = [b,[a,b]] = 0.$$ how do I show that all eigenvalues of $[a,b]$ ...
2
votes
0answers
89 views

Commutator formula in infinite dimensions

The commutator formula states that for $A,B$ elements of a Lie algebra, $$ \lim_{n\to \infty}\left\{ ...
5
votes
0answers
101 views

Quantum Adjoint Action of the Coordinate Algebra on the Enveloping Algebra

As is well known, any Lie group $G$ has a canonical action on its Lie algebra $\frak{g}$, namely the adjoint action $Ad$. Firstly, let me ask, does this extend to an action of $G$ on its enveloping ...
0
votes
0answers
79 views

a question on weyl group and its action on $\mathfrak{t}$

$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{g}l(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the ...
2
votes
2answers
102 views

a question on weyl group

$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{g}l(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the ...
1
vote
1answer
155 views

Derivative wrt. to Lie bracket.

Let $\mathbf{G}$ be a matrix Lie group, $\frak{g}$ the corresponding Lie algebra, $\widehat{\mathbf{x}} = \sum_i^m x_i G_i$ the corresponding hat-operator ($G_i$ the $i$th basis vector of the tangent ...
3
votes
1answer
337 views

Why were Lie algebras called infinitesimal groups?

Why were Lie algebras called infinitesimal groups in the past? And why did mathematicians begin to avoid calling them infinitesimal groups and switch to calling them Lie algebras?
4
votes
1answer
186 views

Representation of Lie algebra of $\textrm{SU}(2)$

$V_m=$Homogeneous polynomials in complex variable with total degree $m$, Let $U\in SU(2)$ is just a linear map on $\mathbb{C}^2$, Define a Linear Transformation $\Pi_m:V_m\rightarrow V_m$ given by ...
1
vote
1answer
99 views

Some representation of $SU(2)$

$V_m=$Homogeneous polynomials in complex variable with total degree $m$, could any one tell me how is that Linear Transformation look like explicitly?And would you please tell me how this is an ...
3
votes
1answer
76 views

correspondence between finite dimensional complex representation

I would like to understand the following fact, shall need help, Thank you. " There is a one- to- one correspondence between the finite dimensional complex representation $\Pi$ of $SU(3)$ and finite ...
1
vote
1answer
158 views

Matrix Lie group counter-example: $e^X$ in the Lie group, but $X$ is not in the Lie algebra

What's an example of a Lie group $G$ and matrix $X$ such that $e^X \in G$ but $x \notin \mathfrak{g}$, where $\mathfrak{g}$ is the associated Lie algebra? This is the same as problem 2.10 in Bryan ...
3
votes
1answer
345 views

understanding adjoint representation

I need to understand what is meant by "tangent space at identity of a Lie group is canonically isomorphic to its Lie algebra" to understand the definition of adjoint representation. Could any one ...
3
votes
1answer
81 views

From $\mathfrak{so}(16)$ to $\mathfrak{su}(11)$?

The two compact real form Lie algebras $\mathfrak{so}(16)$ and $\mathfrak{su}(11)$ have the same dimension (120). They are certainly not isomorphic, but does there exist some kind of algebraic ...
3
votes
0answers
283 views

Fundamental vector fields

I have a question related to fundamental vector fields. For that I first setup the notations and properties etc. Let $G$ be a lie group acting smoothly on the manifold $M$. Let $\mathfrak{g}$ be its ...
1
vote
1answer
107 views

Weyl group of this root system is $S_n$?

Let $E=\{(x_1,\dots,x_n)\in \mathbb{R}^n:\sum x_i=0\}$ is of dimension $n-1$ take root system $R=\{e_1-e_j:1\ge i,j\le n\}$, I know that it is of rank $n-1$ root system, but why its weyl group is ...
0
votes
1answer
77 views

weyl group act by conjugation?

let $W$ be a weyl group and $\alpha\in R$ we have $s_{w(\alpha)}=ws_{\alpha}w^{-1}$, to prove this the author says $ws_{\alpha}w^{-1}$ acts as identity on $wL_{\alpha}=L_{w(\alpha)}$ , and ...
1
vote
1answer
194 views

not all automorphisms of a root system are elements of weyl group

could any one tell me why not all automorphisms of a root system are elements of weyl group? For example in $A_n, n>2$ the automorphism $\alpha\mapsto -\alpha$ is not in the weyl group. I do not ...
8
votes
0answers
117 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
1
vote
1answer
65 views

example of a nilpotent lie algera

I want some example of nilpotent lie algebras, here I also want to see how the set of all $n\times n$ matrices $(a_{ij})$ where $a_{ij} = 0\ \forall\ i\ge j$ forms a nilpotent lie algebra under the ...
1
vote
2answers
49 views

$\alpha\in R$ then if $k\alpha\in R$ then $k=1,2,-1,-2,1/2,-1/2$

Let $R$ be a root system. Suppose $\alpha\in R$ and if for some $k \in \Bbb{R}$ we have $k\alpha\in R$ then how to prove $k=1,2,-1,-2,1/2,-1/2$? I just know ...
1
vote
1answer
187 views

Lie bracket and connection of a surface

Let $f:U \to \mathbb{R}^3$ be a surface where $U \subset \mathbb{R}^2$ is open.Let $\Gamma(Tf)$ denote the space of smooth tangent vector fields on $f$ A connection on $f$ is a map $D:\Gamma(Tf) ...
2
votes
0answers
38 views

Smallest dimensional irreps of semi-simple Lie algebras

I'm wondering if there is a reference that lists the first couple smallest dimensional irreducible representations of each semi-simple Lie algebra. I know these can be found using the Weyl dimension ...
8
votes
3answers
506 views

Lie algebra action from Lie group action: coordinates

Here's the setup: I have $SL(2;\mathbb{C})$ acting on $V = \mathbb{C}[z,w] = \oplus_d V_d$, where $V_d$ is the homogeneous complex polynomials of degree $d$. The action is precomposition: ...
5
votes
1answer
222 views

Methods of Multilinear Algebra in Representation Theory

I have been interested in representation theory lately in particular on that of Lie algebras. Now I have noticed that one way of building representations is to take tensor/exterior/symmetric powers. I ...
1
vote
1answer
110 views

Action of $\mathfrak{sl}_2(\Bbb{C})$ on $\textrm{Sym}^2 V$

I am reading Fulton and Harris and on page 150, there is the following passage (in the second paragraph) that I don't understand: "Similarly, a basis for the symmetric square $W = ...
2
votes
2answers
1k views

Lie algebra of Heisenberg group

To find the Lie algebra of the Heisenberg group $H$, which we know to consist of upper triangular matrices, we see that exponentials of all strictly upper triangular matrices are in $H$. I do not get ...