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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4
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536 views

Relation between root systems and representations of complex semisimple Lie algebras

I'm trying to understand the machinery of root systems for the purpose of classifying complex semisimple Lie algebras. During this process i lost the overview, espacially when it came to highest ...
1
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1answer
80 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 ...
1
vote
1answer
44 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) ...
4
votes
2answers
296 views

Lie algebra isomorphism between ${\rm sl}(2,{\bf C})$ 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 ...
2
votes
2answers
160 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 ...
1
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1answer
94 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 ...
4
votes
3answers
115 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 ...
4
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1answer
92 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 ...
4
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1answer
62 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= ...
2
votes
2answers
426 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 ...
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1answer
311 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 $G$ ...
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0answers
64 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 $M$ ...
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1answer
61 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 ...
3
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0answers
216 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 ...
2
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1answer
219 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 ...
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0answers
33 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 ...
0
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1answer
81 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 ...
4
votes
3answers
129 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
26 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 + ...
4
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1answer
180 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
34 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 ...
3
votes
3answers
224 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
122 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 ...
5
votes
2answers
262 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 ...
2
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0answers
70 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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34 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 ...
3
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0answers
74 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 ...
0
votes
1answer
35 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
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1answer
392 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 ...
4
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1answer
158 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
259 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
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2answers
96 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 ...
2
votes
1answer
168 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 ...
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0answers
46 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
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0answers
70 views

Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)

This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups. More precisely, given discrete groups below (a), (b), (c): ...
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93 views

Definitions of Semisimple Lie Algebra

We usually define semisimplicity of a Lie algebra $\mathfrak{g}\subset M_n ({\bf R})$ from two ways. I want to know the relation between them. One of the definitions of semisimple Lie algebra is ...
3
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1answer
43 views

proving that symplectic lie algebra is a subalgebra of GL

Suppose $S$ is n by n matrix over a field F. Define $gl_S(n,F)=\{A \in gl(n,F): SA+A^TS= 0\}$ Show that this is a subalgebra of $gl(n,F)$ I get as far as: $A \in gl_S(n,F)$ and $B \in gl_S(n,F)$ ...
3
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1answer
106 views

Compactness of semisimple Lie algebra

I want to prove that on a semisimple Lie algebra $\mathfrak{g}$ over ${\bf R}$: $\mathfrak{g}$ is compact if and only if the Killing form is strictly negative definite. Here the Lie algebra is ...
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0answers
611 views

Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
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0answers
170 views

Semi-simple Lie algebra $L$ coincides with its derived algebra $L'$

If $L$ is a semi-simple Lie algebra, then $L=L'$. Since $L$ is semi-simple we can write it as a direct sum of simple ideals $L_i$, i.e. $L=\oplus_{i=1}^r L_i$. Then $L'=\oplus_{i=1}^r L_i'$ and ...
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1answer
88 views

Derivations on semisimple Lie algebra

First recall some definitions : Let $B$ be a Killing form on Lie algebra $\mathfrak{g}$ over ${\bf R}$ such that $B(X,Y)\doteq Tr(ad_Xad_Y)$. $\mathfrak{g}$ is semisimple if $B$ is ...
3
votes
1answer
41 views

Isomorphy of $2$-dimensional $L$-modules

Let $L$ be the $2$-dimensional Lie algebra over $\mathbb C$ with basis $x, y$ and $[xy] = x$. Let $V$ be a vector space over $\mathbb C$ with basis $v_1, v_2$. Then $V$ is an $L$-module if we ...
3
votes
1answer
81 views

Centralizers of connected linear group and its Lie algebra

If we have that $G$ is a connected linear group and $H<G$, where $H$ is also connected, with $\mathfrak{h}$ the lie algebra of $H$ and we define the centralizers of the elements in the following ...
4
votes
2answers
151 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: ...
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votes
2answers
576 views

Calculating the lie algebra of $SO(2,1)$

I am trying to calculate the Lie algebra of the group $SO(2,1)$ where this is defined as: $SO(2,1)=\{X\in Mat_3(\mathbb{R})|X^t\eta X=\eta, \det(X)=1\}$ where $\eta$ is the matrix defined as: $$\left ...
3
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1answer
155 views

The normalizer of a proper sub-algebra properly contains the sub-algebra in a nilpotent Lie algebra.

So I am just making my way into the theory of Lie Algebras. The question at hand comes from page 14 of Humphreys, Introduction to Lie Algebra and Representation Theory. Given a finite dimensional ...
2
votes
1answer
300 views

Are (semi)-simple Lie algebras not solvable?

Let $L\ne \{0\}$ be a non-abelian simple Lie algebra, then the only ideals are $L$ and $\{0\}$. We know that $L'$ is the smallest ideal of $L$ such that $L/L'$ is abelian. If $L'=\{0\}$, then ...
1
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1answer
39 views

Lie subalgebra, Lie subgroup and membership

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $H$ be a connected Lie subgroup with Lie algebra $\mathfrak{h}$. We have that $X \in \mathfrak{h} $ iff $exp(tX) \in H \ \ \ \forall t ...
3
votes
3answers
130 views

Show that $\exp: \mathfrak{sl}(n,\mathbb R)\to \operatorname{SL}(n,\mathbb R)$ is not surjective

It is well known that for $n=2$, this holds. The polar decomposition provides the topology of $\operatorname{SL}(n,\mathbb R)$ as the product of symmetric matrices and orthogonal matrices, which can ...
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0answers
84 views

Proof of Lie theorem on solvable Lie algebra

I am reading a book of Helgason. As you know, solvable Lie algebra $g \subset V= {\bf C}^n$ have a nonzero $v$ such that $v$ is an eigenvector of any element of $g$. I can follow the proof in ...