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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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
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1answer
90 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
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1answer
99 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
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0answers
177 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
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2answers
195 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
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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
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1answer
110 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 ...
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0answers
70 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? ...
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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). ...
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1answer
100 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
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1answer
130 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 ...
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0answers
97 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 ...
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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
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2answers
605 views

Example of two-dimensional non-abelian Lie algebra?

can some one give me an example of two-dimensional non-abelian Lie algebra?
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2answers
767 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} ...
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0answers
53 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
247 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
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1answer
102 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
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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
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0answers
98 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
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0answers
77 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
100 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
151 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
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1answer
299 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
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1answer
181 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
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1answer
98 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
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1answer
155 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
332 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
80 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 ...
2
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0answers
254 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
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1answer
102 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
72 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
163 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
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0answers
114 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
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1answer
62 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
47 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
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1answer
177 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
36 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 ...
7
votes
3answers
485 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
212 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
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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 ...
5
votes
1answer
227 views

History of Lie algebra notation (in Fraktur)?

Does anyone know how it has become the standard to express Lie algebras in fraktur? I'd also like to know how it's established for each era and region, not only the origin. It doesn't seem that ...
2
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1answer
104 views

Irreducibility of a 4 dimensional representation of $\mathfrak{sl}_2(\Bbb{C})$

Let $\mathfrak{g}$ be the complex Lie algebra $\mathfrak{sl}_3(\Bbb{C})$. Consider adjoint representation $\textrm{ad} : \mathfrak{sl}_3(\Bbb{C}) \to \textrm{gl}(\mathfrak{g})$. $\mathfrak{g}$ has ...
5
votes
0answers
77 views

Are there any infinite dimensional subalgebras of the Witt algebra?

The Lie bracket of elements of the Witt algebra is given by: $[L_m,L_n]=(m-n)L_{m+n}$ Are there any infinite dimensional subalgebras of the Witt algebra that are not isomorphic to the Witt algebra ...
3
votes
2answers
192 views

$Z(\mathfrak{g})$ equals Lie algebra of $Z(G)$?

I have tried to prove the following problem: Suppose $G$ is a connected matrix Lie group. Then the center of $\mathfrak{g}$ which is $\ker \textrm{ad}$ is equal to the Lie algebra of $Z(G)$. ...
6
votes
1answer
434 views

Normal Subgroups of Lie groups

I am trying to prove the following. Suppose that $G$ is a matrix Lie group and $H$ a subgroup of $G$. Let $\mathfrak{h},\mathfrak{g}$ be the Lie algebras respectively of $G$ and $H$ with ...
6
votes
0answers
154 views

Generating function for characters of representations

One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ ...
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2answers
899 views

References on Linear Algebraic Groups/Lie Theory

I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the ...