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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Why is 1/2+1/2 in the weight space for SO(5)

Let's consider $\mathfrak{so}(5)$ as the Lie algebra of $\mathrm{SO}(5)$, where the symmetric bilinear form is $x_1y_5+\cdots +y_1x_5$. Then the maximal torus is given by $$\left(\begin{array}{cccccc} ...
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28 views

Moment map and Hamiltonian

Take the manifold $M$ to be $M=\mathbb{R}^6=\mathbb{R}^3\times\mathbb{R}^3$ (hence $x\in M$ is given by $x=(p,q)$ with $p$ and $q$ three dimensional vectors) and take the possion bracket on $M$ given ...
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Exercise 11, chapter 2 in Lie Groups, Lie Algebras, and Representations: An Elementary Introduction

I am reading the book: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction by Brian C. Hall. I am stuck at the following exercise: exercise 11, chapter 2 . Can you help me? ...
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1answer
33 views

Lie algebras with different bases

I am interesting to know that if a finite dimensional Lie algebra $L$ has two bases $\beta_1$ and $\beta_2$, how can we compare the cardinal of two sets $\{(x,y)\in \beta_1\times ...
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1answer
40 views

Embedding so(n) in su(n)

Is there any way of embedding $\mathfrak{so}(n)$ into $\mathfrak{su}(n)$ for any $n$ other than picking the antisymmetric matrices of $\mathfrak{su}(n)$? I know that for small $n$ one can use ...
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27 views

Reducible Lie Algebra

I'm furthering my physics knowledge through a book called Lie Algebras in Particle Physics and am having trouble with one aspect of a problem. I believe because it's a question purely about ...
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1answer
33 views

Representations of the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$

In Corollary 7.2 of http://math.uchicago.edu/~may/REU2012/REUPapers/Bosshardt.pdf, why is the set of weights an unbroken string? I understand we get a finite number of weights by looking at the ...
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1answer
31 views

matrix Lie group embedding as a manifold

Given a Lie group of matrices, and suppose for simplicity that it is globally generated through exponential map from its Lie algebra on a element. Is there a canonical way to embed it into ...
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42 views

Induced Lie Algebra Representation, Left invariant vector fields and more…

The following is an excerpt from a proof in John Lee's Introduction to Smooth Manifolds I am struggling to understand. I would appreciate if someone was able to help me with whatever it is I am ...
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1answer
121 views

Guided study of Lie theory

I'm willing to become a mathematical physicist and, as such, it is mandatory that I become better acquainted with some essential mathematical theories, Lie groups and algebras being one of them. ...
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136 views

When do two matrices have the same exponential?

Let $A$ and $B$ be $n\times n$ hermitean matrices. When do we have $e^{iA}=e^{iB}$? Can we somehow classify those pairs of matrices that have the same exponential? Here are some observations that I ...
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1answer
62 views

kernel of the exponential map is isomorphic to the singular homology group

Let $G$ be an algebraic torus or an abelian variety over the complex numbers. Then $G(\mathbb{C})$ is a complex Lie group. Is it true that we have the following exact sequence ? $ 0 \to ...
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28 views

How to prove this statement for a Lie algebra?

Let $\mathfrak{L}$ be a semi-simple Lie algebra. Let $X^A$ be the elements of this algebra with $A=1, \ldots, N$. The bracket is given by $$[X^A, X^B]=if_{\,\,\,\,\, C}^{AB}X^C$$ where ...
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43 views

Lie bracket equation

I need to solve the equation system of the Lie brackets of vector fields. So I want to find vector fields $X,Y,Z$ such that $F:(\mathbb R^3,\times)\to (V(\mathbb R^3),[.,.])$ , ...
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43 views

Maximal tori in Lie vs algebraic groups

If $G$ is a Lie group, we define a maximal [Lie] torus in $G$ to be a maximal connected compact abelian Lie subgroup of $G$. These guys correspond to Cartan subalgebras of $\mathfrak{g}=Lie(G)$. If ...
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1answer
51 views

How to find the character of $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$?

Let $\mathfrak g$ be a Kac-Moody algebra. Then $$ \mathfrak{n}_{-}=\oplus_{\alpha\in\varPhi_{+}}\mathfrak{g}_{-\alpha} $$ and for $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$ the ...
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2answers
70 views

how to find an integral curve in Lie group?

Given a Lie group $G$, $e$ is its identity element and $g$ is one element of $G$. I want do find a curve $\gamma(t)$ that satisfies these conditions: 1) passes $g$ and $e$, that is ...
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22 views

An injection of Weyl groups

I've shown, quite accidentally, that Weyl group of $F_4$ injects into the Weyl group of $E_6$ as the subgroup of elements normalizing a maximal torus $T^4$ of $F_4$. One might a priori expect other ...
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1answer
70 views

how to extend a vector at $e$ of a Lie group to a left invariant vector field?

I am reading some books about Lie group and Lie algebra. Denote the set of all the left invariant vector fields as $\mathfrak{X}_L$, and the tangent space at $e$ of $G$ as $T_eG$. They say that the ...
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34 views

Chevalley group

Let $L=\mathfrak sl_6$ be the special linear Lie algebra over $\mathbb C$ and let $S=\{\alpha_1, \alpha_2, \alpha_3,\alpha_4,\alpha_5\}$ be a set of simple roots. Then the set of positive roots are ...
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46 views

Fundamental representation of $O(3)$

I want to check if the fundamental representation of $O(3)$ is irreducible on $\mathbb{R}^3$ and $\mathbb{C}^3$. I want to use isomorphism properties. I know this relation exists $$ ...
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51 views

Geodesics on Homogeneous Spaces

Consider a homogeneous space $G/\text{Stab}_p \cong M$ where $G$ is a compact Lie group active transitively on $M$ (a compact manifold). If $F$ is a Finsler Metric on $G$ which pushes forward ...
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1answer
41 views

Why do we have to classify semisimple Lie algebras?

Almost all Lie algebras textbooks deal with the classification of semisimple Lie algebras. Why is it so important?
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24 views

Reference request: Carnot Algebra

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...
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130 views

Reconstructing Lie group globally from the exponential map

This should be an elementary question in Lie group theory, although I'm pretty sure I'm mixing up concepts. Any help clarifying would be much appreciated. Set up Let $G$ be finite-dimensional real ...
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53 views

cohomology ring of abelian Lie algebra

This is a continuation of this question. Let $\mathfrak{g}$ be a Lie $R$-algebra with module basis $e_1,\ldots,e_n$ and zero brackets. Then in the Chevalley (co)chain complex, ...
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cohomology ring of Lie algebras: multiplication

If $\mathfrak{g}$ is a Lie $R$-algebra, then the Chevalley-Eilenberg complex defines the cohomology modules $H^k(\mathfrak{g})$. If $H^\ast(\mathfrak{g})=\bigoplus_kH^k(\mathfrak{g})$, then the ...
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1answer
136 views

Lie algebras of reductive groups

Let $k$ be an algebraically closed field of positive characteristic and let $G$ be a connected split reductive group. We know $G$ is the product of its center $Z(G)$ and derived group $[G, G]$ and ...
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1answer
70 views

How to prove that Lie's theorem fails for a field of prime characteristic?

Lie's theorem say that Let $L$ be a solvable, nonzero subalgebra of $\mathfrak{gl}(V)$, with $V$ a finite dimensional vector space over an algebraically closed field $F$ of characteristic zero. ...
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1answer
65 views

how to calculate a vector in a left invariant vector field?

I would like to understand the left invariant vector field by using a numerical example. Now we consider a Lie group $G=SE(3)$, and the associated Lie algebra is $\mathfrak{g}=se(3)$. We suppose: ...
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15 views

Centre of Lie Algebra $sl_2(\mathbb{F})$

For $L=sl_2(\mathbb{F})$ i.e. matrices with trace zero, what is the centre i.e. $Z(L)$= {$x\in L : [x,y]=[y,x] \ \forall\ y \in L$}. I will have to find matrices $A \in L$ such that $AB=BA$ for all ...
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centre of universal enveloping algebra for specific algebras

Let $\mathfrak{h}_{2n}$ be the Heisenberg Lie algebra, i.e. the Lie algebra with a basis of $\{p_1,\ldots,p_n,q_1,\ldots,q_n,c\}$ where $$[Pi, Pj ] = [Qi, Qj ] = [Pi, C] = [Qi, C] = [C, C] = 0, [Pi, ...
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16 views

defining epimorphism between Lie algebras

It is a general question: Let $A$ and $B$ are two lie algebras over field of characteristic $p>3$ and we have the generators of them. I want to define an epimorphism between them. How we can define ...
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1answer
45 views

how to understand that the set of left invariant vector fileds and $T_eG$ are isomorphic

firstly, the set of all vector fields on $G$ is the subalgebra of left invariant vector fields. I read some reference and find that we do research of algebra always by left invariant, my question is ...
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34 views

cohomology of general linear group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $\mathrm{GL}_n(\mathbb{Z}_2)$ be the group consisting of all $n\times n$ matrices with entries in $\mathbb{Z}_2$ with non-zero determinant. What is the ...
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1answer
52 views

Lie groups and Lie algebras

Ok so I'm confused about the relation between these two concepts. If I have a Lie Group $G$ I can associate a Lie algebra $\mathfrak{g}$ by taking his tangent space in the identity, with the ...
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16 views

Finding coefficients for this Lie algebra isomorphism

This is a question closely related to my previous questions. How, in this thread here did Hee Kwon Lee find the coefficients $(-i,1,0),\ (-i,-1,0)$ and $(0,0,2i)$? In the linked thread ...
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59 views

How to use the Killing form to write down a Lie group isomorphism, and what is the induced Lie algebra isomorphism?

This is a follow up question on my previous question here. One of the answers suggests that I find a map $$SL_2(\mathbb{C}) \to SO_3(\mathbb{C})$$ and then the map induced by this map will be a Lie ...
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1answer
27 views

Solving a large system of linear equations to satisfy the Lie bracket: am I doing it right?

I'm still working on a Lie algebra isomorphism from the Lie algebra of $SL_2(\mathbb C)$ into the Lie algebra of $O(3, \mathbb C)$. It has been suggested to me to use linear combinations of ...
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1answer
55 views

Lie algebra homomorphism: is my understanding correct?

Using answer to my previous question I made some progress towards understanding Lie algebra homomorphisms. But of course I am unsure whether my thoughts are really correct so again I'd like to request ...
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63 views

When did the notion of the tangent space emerge?

When did the modern conception of/notation for the tangent space to a manifold come into use?
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25 views

Importance of Jordan-Chevalley decomposition

What are the uses of Jordan-Chevalley decomposition in the classification of semisimple Lie algebras? I used it to prove that the restriction of the killing form to a maximal toral subalgebra is non ...
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1answer
20 views

Lie subalgebras of $so(3)$ and complex Lie subalgebras of $sl(2,\mathbb{C})$.

I am looking for a reference that: describes all Lie subalgebras of $so(3)$,and describes all complex Lie subalgebras of $sl(2,\mathbb{C})$. Does someone have appropriate references?
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24 views

If $[A,B], A \in \mathcal{L}$ then does this imply that $B \in \mathcal{L}$?

If $[A,B]$ and $A$ are in Lie algebra $\mathcal{L}$ then does this imply that $B \in \mathcal{L}$?
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positive roots remain positive

Let $W$ be the Weyl group of $SL_{n+1}$ and $w \in W$. Let $R^+$ denote the set of positive roots with respect to the Borel subgroup of upper triangular matrices. Define $R^+(w)=\{\alpha \in R^+: ...
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1answer
47 views

Is the exponential map ever not injective?

Let $G\subseteq GL_n(\mathbb R)$ and let $\mathfrak g$ denote its Lie algebra. Let $e: \mathfrak g \to G$ be the map $X \mapsto e^X$. Does there exist an example of $G$ and $\mathfrak g$ such ...
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2answers
189 views

Defining an isomorphism that respects the Lie bracket: is my work correct?

I previously determined that if $\mathfrak{sl}$ denotes the Lie algebra of $SL_2(\mathbb C)$ and $\mathfrak o$ denotes the Lie algebra of $O(3,\mathbb C)$ then a basis for $\mathfrak{sl}$ is given by ...
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17 views

Unimodular Lie group property based on the self-adjoint application

Let $\{e_1, e_2, e_3\}$ a pseudo-orthonormal basis of $\mathcal g$, definined the linear transformation $L:\mathcal g \rightarrow \mathcal g$ such that $L(e_1)=[e_2,e_3]$, $L(e_2)=[e_3,e_1]$, ...
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21 views

Determining whether a Lie algebra is also a complex Lie algebra

I am trying to learn Lie theory. In the following I will share my thoughts. Please, can you check my work for correctness and point out any mistakes to me? I am trying to determine whether ...
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2answers
22 views

Center of compact lie group closed?

Let me specify that my knowledge about Lie groups/algebras is reduced to bits and pieces I learned from various diff geometry textbooks. I could not find a reference for the following question (I am ...