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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14 views
Weight spaces of Verma modules
Let $\mathfrak g$ be a semisimple Lie algebra generated by $x_i^+,x_i^-$, $1\leq i\leq n$, via the Chevalley-Serre relations and let $V(\mu)$ be a Verma module with highest-weight $\mu$. I gather that ...
4
votes
1answer
124 views
Dynkin diagram automorphisms and weights
Let $\sigma$ be a nontrivial Dynkin diagram automorphism of a finite-dimensional complex simple Lie algebra $\frak g$ (of type A, D or E) and let $\frak h$ be a Cartan subalgebra of $\frak g$. Let $I$ ...
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0answers
12 views
relation between the Poincaré and Euclidean algebra
Take $d$ a strictly positive integer, and consider the (proper) Euclidean group $E^d$ (the symmetry group of $\mathbf{R}^d$ with the conventional inner product), and the (proper, ortochronous) ...
3
votes
1answer
104 views
What is the difference between $\ker( L \bigwedge L \overset{[-,-]}{\rightarrow} L )$ and $\langle a \wedge b \big| [a,b]=0\rangle$?
Let $L$ be a finite dimensional Lie algebra.
We view the Lie bracket as a linear map on the exterior square:
$$\pi:L \bigwedge L \rightarrow L$$
Define $$\bigwedge L := \langle a \wedge b \big| ...
1
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1answer
31 views
Lie algebra isomorphism between $sl(2,{\bf C})$ and ${\bf C}^3$
I think that this is an exercise. I can not find a solution.
We can define Lie bracket multiplication on ${\bf C}^3$ : $$ x\wedge y $$ where $x=(x_1,x_2, x_3)$, $y=
(y_1,y_2,y_3)$, and $\wedge $ is ...
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0answers
51 views
Integer conjugacy class
I am wondering about the following thing: What are integer conjugacy classes? Could anybody please give me a definition and maybe one or two examples?
What is meant with an integer conjugacy class ...
3
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0answers
30 views
Root Decomposition on Semisimple Lie Algebra over ${\bf C}$
Let $\mathfrak{g}$ is a semisimple Lie algebra over ${\bf C}$.
Then we have a direct sum $$ \mathfrak{g} = \mathfrak{h} + \sum_{\alpha}
\mathfrak{g}^\alpha $$.
where $\mathfrak{h}$ is a Cartan ...
8
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1answer
96 views
Why is the Lie derivative linear in the vector field?
This might seem a very basic question, but I can't manage to find a proper proof in the books I have on my desk (or simply cannot see that it's "just that"). So be sure of what we talk about, let $G$ ...
2
votes
2answers
37 views
Enveloping Algebra $U(L \oplus L')$
I'm having trouble understanding part of a proof of the following statement
Let $L,L'$ be Lie algebras and $L \oplus L'$ their direct sum. Then $$ U(L \oplus L') \cong U(L) \otimes U(L')$$
Let ...
4
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1answer
14 views
Infinitesimal $SO(N)$ transformations
An infinitesimal $SO(N)$ transformation matrix can be written :
$$R_{ij} = \delta_{ij}+\theta_{ij}+O(\theta^2)$$
Now it has to be shown that $\theta_{ij}$ is real and anti-symmetric.
I've started ...
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1answer
60 views
$\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group?
I try to prove $\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group.
My idea is that since $\exp\colon \mathfrak{sl}(2,\mathbb{R}) \to ...
3
votes
3answers
84 views
What does boson-type realization mean?
I have seen several different contexts the expression "boson-type realization", for instance in the study of algebras growth and realization of affine algebras.
To be or not be a boson-type ...
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1answer
19 views
How to write down a presentation of a Lie algebra if we know a set of generators?
How to write down a presentation of a Lie algebra if we know a set of generators in matrix form? For example, for $sl_2$, if we know $e=(0, 1; 0, 0)$, $f=(0, 0; 1, 0)$ , $h=(1, 0; 0, -1)$, how to ...
2
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2answers
80 views
Weyl group of the Lie algebra $\mathfrak{sl}_n$
The Weyl group of the Lie algebra $\mathfrak{sl}_n$ is just the symmetric group on $n$ elements, $S_n$. The action can be realized as follows. If $\mathfrak{h}$ is the Cartan subalgebra of all ...
4
votes
1answer
39 views
solvable subalgebra
I want to show that a set $B\subset L$ is a maximal solvable subalgebra.
With $L = \mathscr{o}(8,F)$, $F$ and algebraically closed field, and $\operatorname{char}(F)=0$ and
$$B= ...
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1answer
45 views
Example ideal of $\mathfrak{sl}(2,\mathbb{C})$
I need an example about ideal from lie algebra $\mathfrak{sl}(2,\mathbb{C})$ except trivial ideal and $\mathfrak{sl}(2,\mathbb{C})$ itself, can someone help me?
I try to make ideal except trivial ...
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votes
1answer
41 views
Classifying all rank 2 and 3 root systems
I am working with the representation theory of complex simple Lie algebras, and have a question:
It is intuitively clear that the root systems $A_1\times A_1$, $A_2$, $B_2$, and $G_2$ comprise all ...
2
votes
0answers
54 views
Finite-dimensional Lie algebra as a scheme
Kindly asking for any hints about the following questions:
Suppose $k$ is an algebraically closed field of characteristic zero and $g$ is a finite-dimensional Lie algebra over $k$. Then $g$ is ...
0
votes
1answer
127 views
How to use Weyl dimension formula?
Let $V(\lambda)$ be a highest weight module of a semi-simple Lie algebra with highest weight $\lambda$. The Weyl dimension formula is $\dim V(\lambda) = \frac{\prod_{\alpha>0} (\lambda+\rho, ...
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0answers
16 views
Root space decomposition of semisimple Lie algebras
Let $H \subset L$ be a Cartan subalgebra of a semisimple - and in particular, finite dimensional - complex Lie algebra $L$.
Then $\operatorname{ad}(H) \subset \operatorname{End}_\mathbb{C}(L)$ ...
4
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1answer
56 views
Exercise in Erdmann's Intro to Lie algebras
I'm working on question 4.8 on page 36 of Erdmann's book called Introduction to Lie Algebras. The question is as follows:
Let $L$ be a Lie algebra over a field $F$, such that $[a,b],b]=0$ for all ...
4
votes
2answers
115 views
path-connected subgroup of Lie group is Lie group
Theorem (Yamabe): Let $G$ be a Lie group, and let $H$ be an arc-wise connected subgroup of $G$.Then $H$ is a Lie subgroup of $G$.
I am reading this theory form an appendix in a book called (bilinear ...
3
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0answers
54 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 ...
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0answers
40 views
Heisenberg algebra and other Lie algberas
Is there a sub Lie algebra $K$ such that for an ideal $M$ of a heisenberg algebra $H$,
$H=K+M$ and $K\cap M=0$ ($M$ has a complement in $H$)?
Is there a class of Lie algebras such every ideal $M$ ...
4
votes
1answer
65 views
The set of complete vector fields
The set of all complete vector fields in $\mathbb R^{n}$ is closed under Lie bracket? is this set a $D$-module where $D$ is the ring of bounded smooth funcions? Can anyone recomend me a book on the ...
1
vote
1answer
40 views
if $X$ is a vector field how can I find $Y$ such that $[X,Y]=0$?
Suppose I am given a holomorphic vector field $X$ over a complex manifold $M$. To simplify this we can suppose that $X$ is a holomorphic vector field in $\mathbb{C}^n$ for $n=2$ or $n=3$. How can I ...
2
votes
0answers
74 views
left-invariant n-form and metric on a Lie group
These two questions are from my exam practice question sets , which are quite similar. I got some problem understanding and solving both of them .
For (a) , I can only substite $dx\wedge dy\wedge ...
1
vote
0answers
26 views
Representation of Complexification of Lie Algebra
Is the following obvious? I think it is, but wanted to make sure before an exam tomorrow!
"There is a bijection between the complex representations of a real Lie algebra and the complex ...
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0answers
17 views
If Lie(H) preserves a subspace, must H also preserve that subspace?
Assume $H \subset G$ is a closed connected subgroup of a linear algebraic group over an arbitrary field (both assumed to be smooth). Assume $G$ acts linearly on the (finite dimensional) vector space ...
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0answers
28 views
show that if $\beta + n\alpha $ is a root for some integer $n$, then $\beta + n\alpha $ lies in the alpha string through beta.
So I would like to show the following, which is,
If $\beta + n\alpha $ is a root for some integer $n$, then $\beta + n\alpha $ lies in the alpha string through $\beta$.
I'm guessing the fact that if ...
3
votes
3answers
42 views
The Lie algebra of the generators of the projective transformation is isomorphic to the Lie algebra of traceless matrices.
The general projective transformation of the $x$-$y$ plane is given by
$$\tilde{x}=\frac{a_1x+a_2 y+a_3}{a_7x+a_8y+a_9},\quad\tilde{y}=\frac{a_4x+a_5y+a_6}{a_7x+a_8y+a_9}$$
for some constants ...
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0answers
23 views
Closed formula for the coefficients of a series obtained from an expansion.
The Heisenberg algebra is generated by $h_i, i\in \mathbb{Z}\backslash\{0\}$ and the central element $c$. We expand the function
$$\exp (\sum_{n=1}^{\infty}h_{-n}\frac{z^n}{n}) = 1 + ...
3
votes
1answer
42 views
Semisimple complex Lie Algebra and decomposition into weight spaces.
So I was wondering why a semisimple complex Lie Algebra $L$ is a direct sum of its weight spaces.
Given a Cartan Subalgebra of $H$ of $L$ then since $L$ is a semisimple complex Lie Algebra, then ...
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0answers
15 views
Classification of 6-D Nilmanifolds
I am reading the G.Cavalcanti and M.Gualtieri's Generalized Complex Structures on Nilmanifolds. In the introduction it is said that there are 34 nilpotent lie algebra isomorphism classes. There are ...
2
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3answers
67 views
Definition of Lie Algebra
I have just start studying Lie Algebra. I want to know the motivation for the conditions on the Lie Bracket in the Definition of Lie Algebra. Please explain me or tell me some references. Thanks.
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0answers
22 views
A complex lie algebra is the direct sum of simple ideals iff it is semisimple
So I am wanting to show that a complex lie algebra is the direct sum of simple ideals iff it is semisimple.
In fact I have already proved <=
It remains for me to prove =>
$\textbf{Currently I ...
4
votes
2answers
67 views
SO(2,1) not connected
I am trying to show that $SO(2,1)$ is not connected but I have no idea where to start really, I know that it is connected if there is a path between any two points. My definition of $SO(2,1)$ is:
...
3
votes
0answers
59 views
Weights set spans
Definition
Let $T$ be a torus and $R: G \to GL(V)$ a representation. $R(T)$ is a collection of commuting matrices and therefore can be simultaniously diagonalized.
For a character $\lambda \in ...
3
votes
0answers
34 views
Cartan's Criterion. $L$ solvable $\implies$ $Tr(xy) = 0$ $\forall x \in L$, $\forall x \in L^{(1)}$
Cartan's Criterion.
Given $V$ a finite dimensional complex vector space and $L$ a Lie subalgebra of $gl(V)$ then, $L$ solvable $\implies$ $Tr(xy) = 0$ $\forall x \in L$, $\forall x \in L^{(1)}$.
...
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0answers
37 views
Lie Algebras : Showing L is nilpotent if every maximal Lie subalgebra of L is an ideal.
Given a finite dimensional Lie algebra $L$, suppose that each maximal lie subalgebra of $L$ is an ideal. Suppose the adjoint map, $ad_y$ is not nilpotent.
Then pick a maximal subalgebra $M$ ...
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0answers
24 views
Given $L$ a complex finite dimensional Lie algebra. Then suppose $L$ is solvable. Show $L^{(1)}$ is nilpotent.
Given $L$ a complex finite dimensional Lie algebra. Then suppose $L$ is solvable. Show $L^{(1)}$ is nilpotent.
Okay, so I have the existence of a flag of ideas in $L$. Can I deduce from this that ...
1
vote
1answer
139 views
Linear independency of a set of functions.
Let $m\in \mathbb{Z}, \mu_{m}^{j}\in \mathbb{C}, \lambda_{m'}^{j}\in \mathbb{C}, \Psi_{i,r}^{+}\in \mathbb{C}$. $$\lambda^{j}(z)=\sum_{m'\in \mathbb{Z}}\lambda_{m'}^{j}z^{m'}$$ ...
0
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1answer
22 views
Ideals of quotient algebras.
Suppose $I$ and $J$ are ideals of a Lie Algebra L.
I know that we have the fact that:
$\frac{I+J}{J} \cong \frac{I}{I\cap J}$
Prove that the ideals of $\frac{L}{I}$ - the quotient algebra of L ...
2
votes
1answer
46 views
First Homology Group and Abelianization
On the Wolfram Mathworld article for Commutator Subgroup, it states that the first homology group is the abelianization, $$H_{1}(G) = G \big/ [G,G]$$
which totally blows my mind because I've only seen ...
0
votes
0answers
43 views
Lie algebra of Euclidean group
From the book "Spinning Tops" by Audin, she claims that
$$\mathfrak{so}(3)[\epsilon]/\epsilon^2$$
with coefficientwise Lie bracket is a Lie algebra of a Lie group that is $TSO(3)$ (group action not ...
6
votes
2answers
75 views
$V=\{A\in M_{3\times 3}(\mathbb{R}):\text{trace}(A)=0\}$ is isomorphic to $\text{span}\{AB-BA:A,B\in V\}$
Background: Let
$$V:=\{A\in M_{3\times 3}(\mathbb{R}):\text{trace}(A)=0\}$$
be the vector space of $3\times 3$ real matrices with vanishing trace, and let $[\cdot,\cdot]:V\times V\to V$ be defined by
...
0
votes
1answer
60 views
Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?
At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint?
(..a related query: ...
1
vote
1answer
37 views
Show that an Ehresmann connection on a principal G bundle is equivalent to a Lie Algebra Valued one form.
Let $E$ be a smooth principal $G$-bundle on M.
The vertical bundle $V$ is defined as $V=\ker(d\pi:TE\to \pi^*TM)$. An Ehresmann connection on $E$ is a smooth subbundle $H$ of $TE$ (also called the ...
2
votes
0answers
21 views
Prove that $[L,Rad(L)] \subseteq N$ for finite-dimensional Lie algebra $L$
I need to prove the following fact: if $L$ is a finite-dimensional Lie algebra over field of characteristic $0$, $Rad(L)$ is its radical, and $N$ is the maximal nilpotent ideal in $L$, then ...
2
votes
1answer
43 views
Extension of a semisimple Lie algebra with a semisimple Lie algebra is semisimple. Why?
I am looking for a proof, that the extension of a semisimple Lie algebra with a semisimple is again semisimple.
I know the Theorem of Weyl and the one of Levi, so I thought about maybe using them.
