1
vote
2answers
39 views

The discription of abelian Lie groups

There is a problem in my problem sheet to discribe all abelian connected Lie groups (moreover this is the first problem and it should be rahter easy). First it is difficult to understand how this ...
1
vote
0answers
39 views

Exercise in Lie algebra course

Let $A$ and $B$ be subalgebras of a Lie algebra $L$ such that $B\subset N_L(A)$. (a) Verify that the space $A+B$ is a subalgebra of $L$. (b) Verify that $A\triangleleft(A+B)$ and $(A\cap ...
0
votes
0answers
22 views

Constructing Lie algebra from the associative algebra.

Show that any associative algebra $A$ can be made into Lie algebra by taking $[x,y]=xy-yx$ for any $x,y \in A$. The way I would tackle it. $\circ$ Clearly $A$ is a vector space as it is an ...
1
vote
1answer
81 views

Problem 9.7 - Lie Algebras - Humphreys

Let $\alpha,\beta\in\Phi$ span a subspace $E'$ of $E$. Prove that $E'\cap\Phi$ is a root system in $E'$. Prove similarly that $\Phi\cap(\mathbb{Z}\alpha+\mathbb{Z}\beta)$ is a root system in E' (must ...
2
votes
1answer
46 views

Equation On Root Systems (Humphreys Exercise 9.10)

I am stuck in the following problem from Humphreys. Let $\alpha, \beta$ be roots in a root system $\Phi$. Let the $\alpha$-string through $\beta$ be $\beta - r\alpha, \ldots, \beta + q\alpha$ and let ...
2
votes
2answers
171 views

Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV

Let V be a real n-dimensional vector space. Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV. Note that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is a real vector space and is ...
2
votes
0answers
61 views

Complete reducibility of tensor product

Let $L$ be a Lie algebra (over a algebraically closed field, not sure if it is relevant). If $V$ and $W$ are two completely reducible $L$-modules, can anyone give a hint on how to show that $V\otimes ...
1
vote
0answers
43 views

Casimir Invariants within the universal enveloping algebra

I've been asked to determine the eigenvalue of the Casimir invariant $I_2$ on any irreducible module with highest weight $\lambda = (\lambda_1, \lambda_2, ..., \lambda_n)$, where; $$I_m = ...
2
votes
0answers
63 views

Does the Lie Bracket automatically exist?

Let $g$ be a Matrix Lie Group. The Lie Algebra of $g := Lie(g)$ is defined as $ Lie(g) = \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1, \gamma(0) = \mathbb{I} \} $ ...
4
votes
1answer
37 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 ...
0
votes
1answer
71 views

Find a $1$-form $ω$ on $\mathbb R^2 −\{(0,0)\}$ such that $ω(X) = 1$ and $ω(Y) = 0$.

Please ı dont know what I need to do. thus, help me to solve.
2
votes
1answer
298 views

Tangent space at the identity element of a lie group

Let G be a lie group . we know a Lie group is a group with a smooth manifold structure s.t both the multiplication map $m$ and group inversion map $i$ are smooth . Now by identifying ...
0
votes
1answer
67 views

Lie bracket of vector fields on $\Bbb R^{n}$

Please show how to solve? I am stack with lie bracket. Thank you.
0
votes
0answers
46 views

Map algebras between scheme and Lie algebra

Kindly asking for any hints about the following questions: Suppose, $X$ be an arbitrary scheme over an algebraically closed field $k$. 1- In general, what is the structure of $A= ...
1
vote
1answer
52 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 ...
6
votes
0answers
85 views

$\Delta \subset \Phi$ is a base in a root system imples $\Delta^\vee \subset \Phi^\vee$ is a base in a root system

(the notation here is compatible with J.E. Humphrey's "Introduction to Lie Algebras and Representation Theory") Let $\Phi \subset E$ be a root system. Let $\Delta \subset \Phi$ be a base. I already ...
1
vote
1answer
66 views

Lie algebra 3 Dimensional with 2 Dimensional derived lie algebra #2

I read the book of Karin Erdmann and Mark Wildon: "An introduction to Lie algebras". In page 22 they say that: If $\dim (L) = 3$, $\dim (L') = 2$ then (a) $L'$ abelian and (b) $\operatorname{ad} x ...
4
votes
2answers
173 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 ...
-2
votes
1answer
92 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 ...
6
votes
3answers
362 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
2answers
208 views

Subgroups of $\Bbb{R}^n$ that are closed and discrete

I am trying to prove that every closed discrete subgroup of $\Bbb{R}^n$ under addition is a free abelian group of finite rank. I have tried to do this by induction on the dimension $n$. Base ...
12
votes
1answer
586 views

Exercise 6.5 in Humphrey's Book on Lie Algebras

I am trying to solve Exercise 6.5 part 4 in James Humphreys' Introduction to Lie Algebras and Representation Theory. I added the (homework) tag because my question is about an exercise, but this is ...
2
votes
0answers
51 views

ADE type root lattice

Let $\Phi$ be a root system of ADE type, $L$ is the corresponding root lattice, show that $\Phi=\{\alpha\in L:(\alpha,\alpha)=2\}$, where $(,)$ is the normalized non-degenerate symmetric bilinear form ...
1
vote
0answers
38 views

Inverse boson operator realization of $\mathfrak{so}(3)$

This is actually a homework problem. The inverse boson operators $a^{-1}$ and $\left(a^\dagger\right)^{-1}$ are defined as $$a^{-1} |n\rangle = \frac{1}{\sqrt{n+1}} |n+1\rangle$$ ...
2
votes
1answer
77 views

Isomorphism between $sl_{4}$ and the orthogonal group of $6$ variables

Let V be the irreducible $sl_{4}$-module with highest weight $\pi_{2}=\lambda_{1}+\lambda_{2}$ (i.e if $H=\left(\lambda_{1},\dots,\lambda_{4}\right)$ is a diagonal matrix in $sl_{4}$ with values ...
5
votes
2answers
617 views

symplectic lie algebra is simple

The symplectic lie algebra defined by $sp\left(n\right)=\left\{ X\in gl_{2n}\,|\, X^{t}J+JX=0\right\}$ when $J=\begin{pmatrix}0 & I\\ -I & 0\end{pmatrix}$. So $X\in sp\left(n\right)$ is of ...
7
votes
1answer
206 views

Isomorphism of an irreducible module of a certain Lie-algebra

While preparing for a test I found the next question which i cannot fully answer: Assume $k$ is an algebraically closed field, and $g_{1},g_{2}$ are $k$-Lie algebras and let $g=g_{1}\times g_{2}$. ...
2
votes
1answer
97 views

Lie algebras homeomorphism problem

In my homework problem I have to prove that $F: (\Bbb{R}^3,\times) \to (so(3),[,]),\ F(v)=\begin{pmatrix}0&-v_3&v_2 \\ v_3& 0&-v_1\\ -v_2&v_1&0 \end{pmatrix}=\hat{v}$ is a ...
2
votes
2answers
75 views

A solvable Lie-algebra of derived length 2 and nilpotency class $n$

Given a natural $n>2$, I want to show that there exists a lie algebra $g$ which is solvable of derived length 2, but nilpotent of degree $n$. I have seen a parallel idea in groups, but i can't see ...
7
votes
1answer
152 views

Simplicity of $\operatorname{Der} \left(\mathbb F_p [x_1, \dots, x_n]/ (x_1^p,\dots, x_n^p )\right)$

I need to prove that the Lie algebra defined as: $W_{n} = \operatorname{Der} \left(\mathbb F_p [x_1, \dots, x_n ] / (x_1^p, \dots, x_n^p )\right)$, when $(x_1^p, \dots, x_n^p )$ is the ideal generated ...
1
vote
1answer
182 views

The universal enveloping algebra of the trivial lie algebra

Given a commutative ring $k$ and a $k$-lie algebra $g$, I need to prove that the universal enveloping algebra $\mathcal{U}\left(g\right)=k\iff g=\left\{ 0\right\}$. One direction is very easy: If ...
1
vote
1answer
77 views

counter examples for alleged sub-algebras

This is a homework question, in which we've got a bunch of kinds of subsets of a given Lie-algebra, and needed to decide wether these are sub-algebras, ideals, or non of the above. I have managed to ...
3
votes
1answer
199 views

fundamental representation of $\ sl(l+1,F)$

This problem concerns the topic representation theory of Lie Algebras. The main purpose of the exercise is to study the form of the fundamental dominant weights of a Lie Algebra. I would be very ...
1
vote
1answer
124 views

Connection 1-form on Lie group

If we regard $S^{2n-1} \to \mathbb{CP}^{n-1}$ as a principal $S^1$ bundle, how do I show that $$A=\frac{1}{2\pi}\sum_i(x_i dx_i-y_i dy_i),$$ where $(x_1,y_1,\dotsc,x_{2n},y_{2n})$ are coordinates on ...
1
vote
0answers
229 views

How to show that the structure constant of SU(3) is invariant?

So suppose $f_{ijk}$ is the antisymmetric structure constant of SU(3), and $D^8_{ij}(g)$ is the matrices of 8-dimensional adjoint representation of SU(3), then how to show that ...
1
vote
1answer
180 views

Finding roots for a Lie algebra g, wrt toral subalgebra h

I'm trying to find the root space decomposition of a lie algebra wrt a toral subalgebra h. Both a matrix lie algebras. I'm confused about how do I find the linear forms $\lambda \in \mathfrak{h}^*$ ...