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### 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 ...
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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; $$I_m = ... 0answers 63 views ### Does the Lie Bracket automatically exist? Let g be a Matrix Lie Group. The Lie Algebra of g := Lie(g) is defined as  Lie(g) = \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1, \gamma(0) = \mathbb{I} \}  ... 1answer 37 views ### Infinitesimal SO(N) transformations An infinitesimal SO(N) transformation matrix can be written :$$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 ... 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. 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 ... 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. 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= ... 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 ... 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 ... 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 ... 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 ... 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 ... 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: ... 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 ... 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 ... 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 ... 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$$... 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 ... 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 ... 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}. ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ...
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 ...
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 $\lambda \in \mathfrak{h}^*$ ...