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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question on roots and root vectors of a simple lie algebra

Assuming that for each root α there is only one linearly independent root vector, show that if $\alpha$, $\beta$, and $\alpha+\beta$ are roots, then [$e_\alpha$ , $e_\beta$ ] not equal to 0. Here ...
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1answer
84 views

If $\alpha$ is a root of a simple lie algebra, then prove that the only multiples of $\alpha$ which are roots are $\alpha, -\alpha,0$

If $\alpha$ is a root, then the only multiples of α which are roots are $\alpha, -\alpha, 0$. Here $\alpha$ is a root of a simple lie algebra. How do I prove this?
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1answer
62 views

If $\alpha$ is a root of a simple lie algebra, prove that $\langle \alpha,\alpha \rangle \neq 0$

If $\alpha$ is a root of a simple lie algebra, prove that $\langle \alpha,\alpha \rangle$ not equal to $0$. From this, I want to prove that the $\langle,\rangle$ could be used as a scalar product.
3
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1answer
269 views

Conditions for left-invariant one-forms to be closed.

Let $G$ be a connected (semisimple) Lie group with Lie algebra $\frak{g}$. For $\omega \in \frak{g}^*$, we may define a left invariant one-form on $G$ by $\left[ \omega (g)\right] (v)=\omega \left( ...
3
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1answer
137 views

How do I find all the roots of a lie algebra and hence its root system diagram given the cartan matrix for that algebra?

If I am given the cartan matrix, I can find the $2\langle αi , αj\rangle/\langle αj , αj\rangle$ of the simple roots where $α$ are the simple roots. But, from this how do I find the $\langle αi , ...
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0answers
50 views

$G_1$-Scalar factors for Clebsch-Gordan coefficients for $ U(n)$

when evaluating the $G_1$ scalar factors for CGC's of $U(n)$ it seems that some of the factors are undefined. The explicit formula for the evaluation of the scalar factors is Eq. (6) in 18.2.8 of N.J. ...
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1answer
87 views

Question on induction proof (about direct sums of irreducible submodules)

Let $V$ be an $L$-module. I want to show that $V$ is a direct sum of irreducible $L$-submodules if each $L$-submodule of $V$ possesses a complement. I want to show this via induction on the dimension ...
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2answers
262 views

Proving a Lie algebra is simple

Let L be a 3-dimensional vector space over k with basis x,y,z. Given L an anti-commutative algebra structure by setting $[x,y]=z,[y,z]=x,[z,x]=y$ Prove that L is a simple Lie algebra. So L is ...
5
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1answer
423 views

How to prove that a subalgebra $\mathfrak p$ in a semisimple lie algebra $\mathfrak g$ is parabolic if $\mathfrak p^\perp=\mathfrak{rn}(\mathfrak p)$?

Let $\mathfrak g$ be a semissimple Lie algebra over an algebraically closed field of characteristic zero. Suppose $\mathfrak p$ is a subalgebra of $\mathfrak g$ such that $\mathfrak ...
2
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0answers
77 views

Principal Bundles and Lie algebras

Suppose I have a principal $S^{1}$ bundle over a nice compact symmetric space. The symmetric space arises as a homogeneous space, call it $X=G/H$. On the Lie algebra level we have the decomposition ...
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5answers
3k views

Jacobi identity - intuitive explanation

I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can't see through it. I can't see the motivation behind it (as ...
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1answer
627 views

Basis for adjoint representation of $sl(2,F)$

Consider the lie algebra $sl(2,F)$ with standard basis $x=\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$, $j=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, $h=\begin{bmatrix} 1 & 0 ...
2
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2answers
43 views

Showing $R(G) = R(T)^W$

Let $G$ be a compact connected Lie group and $T$ a maximal torus. Let $R(G)$ be the representation ring of $G$. Then restriction of reps gives a map $R(G) \to R(T)^W$, where $R(T)^W$ are the ...
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0answers
124 views

First-order derivatives in differential forms calculus

Let $d$ denote the Cartan differential, and let $\delta$ denote the codifferential. The underlying domain is not important for what follows. The canonical generalization of the Laplace-operator ...
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0answers
316 views

Abelian Cartan subalgebras

If a Lie algebra is semisimple or reductive, its Cartan subalgebras are Abelian, and their elements semisimple. Are there non-reductive algebras with Abelian Cartan subalgebras all of whose ...
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2answers
33 views

Understanding the Lie algebra $o_{V,B}$

I am learning about Lie algebras and I do not understand the following subalgebra of $\mathfrak gl_{V}$. Let $V$ be a vector space and $\mathfrak gl_{V}$ be the Lie algebra of endomorphisms on $V$. ...
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0answers
37 views

Subspace of a homogeneous space.

Given two homegenous spaces $\frac{G}{H}$, $\frac{A}{B}$ with $A\subset B$ is there a way to prove that $\frac{A}{B}\subset \frac{G}{H}$ ie that $B\subset A\cap H$? In particular I would like to ...
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2answers
586 views

Canonical isomorphism between $\mathfrak{so}(3)$ and $\mathbb R^3$ with vector cross product

There is a well-known isomorphism between the Lie algebra $\mathfrak{so}(3)$ and $\mathbb{R}^3$ which maps the Lie bracket to the vector cross product. It looks like $$ \begin{pmatrix} 0 & -z ...
2
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0answers
43 views

Solve commutator relation $[Q,d]=-[P,d]$ for $Q$ on chain complexes with scalar product

Suppose we are given chain sequences $\dots \rightarrow C_k \rightarrow C_{k+1} \rightarrow \dots$ and $\dots \rightarrow D_k \rightarrow D_{k+1} \rightarrow \dots$ of finite-dimensional vector ...
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2answers
710 views

Cayley Transform, Exponential Mapping and more…

Assume a self-adjoint operator, represented as hermitian matrix $H=H^\dagger$. To my knowledge there are at least 2 mappings of $H$ onto unitary matrices: Cayley's Transformation with ...
2
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1answer
80 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 ...
4
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2answers
342 views

What physical meaning do the dimension of Wigner d-matrices have?

Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also ...
3
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1answer
53 views

$\hom_{k}\left(V_{p,q},V_{r,s}\right)\simeq V_{q+r,p+s}$

Define $V_{p,q}=\underset{p}{\underbrace{V\otimes\cdots\otimes V}}\otimes\underset{q}{\underbrace{V^{*}\otimes\cdots\otimes V^{*}}}$. In a previous question here I was shown that ...
6
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2answers
888 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 ...
3
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1answer
50 views

What is the systematic way to convert any arbitrary finite dimensional representation into block diagonal form?

Given any arbitrary representation, how do I convert it into block diagonal form, or find its irreducible representation?
4
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1answer
657 views

Endomorphisms of $V$ and the dual space

I was told that $V\otimes V^{*}\simeq\mbox{End}\left(V\right)$. I can't find the isomorphism itself though. Can anyone tell me what it is with a proof? Thanks!
2
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1answer
83 views

Dimension of Lie algebra

Let $o(2l,F)$, with $l \ge2$ and $n=2l$ be the orthogonal lie algebra $\{L\in gl(n,F)|SL=-L^{t}S\}$ where $S=\begin{pmatrix} 0 &I_{l} \\ I_{l} & 0 \end{pmatrix}$. How can I show that ...
2
votes
1answer
163 views

Linear independency of a set of functions.

Let $m\in \mathbb{Z}, \mu_{m}^{j}\in \mathbb{C}, \lambda_{m'}^{j}\in \mathbb{C}, \Psi_{i,r}^{+}\in \mathbb{C}$. $$\lambda^{j}(z)=\sum_{m'\in \mathbb{Z}}\lambda_{m'}^{j}z^{m'}$$ ...
7
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1answer
210 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 $g=g_{1}\times g_{2}$. ...
4
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0answers
169 views

internal direct product of lie groups

If $G$ is a (edit: simply connected)Lie group, when does a direct sum decomposition of its Lie algebra (into a direct sum of subalgebras) correspond to a (semi)direct product decomposition of $G$? ...
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2answers
86 views

Name of specific solvable Lie Algebra of dimension 4

I have a Lie algebra comprised of the generators $\{e_1,e_2,e_3,e_4\}$ for which the only non-zero commutators are $$ [e_4,e_2]=-i e_3 $$ $$ [e_4,e_3]= i e_2 $$ (Excuse the physicist notation, for ...
2
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1answer
99 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 ...
1
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0answers
116 views

PBW Theorem applied to graded Lie algebras

Fix a $\mathbb Z_+^n$-graded Lie algebra ${\frak a}=\oplus_{r \in\mathbb Z_+^n}^{} {\frak a}[r]$ such that ${\frak g}:={\frak a}[0]$ is a finite-dimensional semisimple Lie algebra over the complex ...
0
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1answer
75 views

question on $\mathfrak{su}(2)$

Let $\sigma_x$, $\sigma_y$, $\sigma_z$ be the standard Pauli matrices. Prove, if $\alpha \cdot \sigma = \alpha_x \sigma_x + \alpha_y \sigma_y + \alpha_z \sigma_z$, that $\alpha \cdot (\sigma \beta) ...
5
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1answer
1k views

Question on fundamental weights and representations

I am a bit confused about the notion of "fundamental weights". In a complexified setting, I am thinking of my Lie algebra to be decomposed as, $\cal{g} = \cal{t} \oplus _\alpha \cal{g}_\alpha$ where ...
4
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2answers
190 views

A trivial question concerning $sl_{n}\mathbb{C}$ representations

The question is, does the fact $$ \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0& 0\\ 0 & 1 &0 \end{array}\right)^{2}=0, \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 ...
1
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1answer
74 views

Action of $L$ on $End(V)$.

I'm reading Introduction ot Lie Algebras and Representation Theory from James Humphreys and I do not understand the statement made at the top of page 27. Given a vectorspace $V$ (finite dimensional) ...
3
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1answer
194 views

Lie algebras and integral curves

I am trying to understand the proof of the following which comes from "Matrix Groups for Undergraduates" by Kristopher Tapp. Let $G \subset GL_n(\mathbb K)$ be a matrix group with Lie algebra ...
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1answer
216 views

Lie Brackets of Nilpotent Lie Algebras

Suppose I have the Heisenberg group H say over the $p$-adic integers $\mathbb{Z}_p$, which is the set of $3\times 3$ uni-upper-trianglar matrices over $\mathbb{Z}_p$ . Its Lie algebra $h$ is the set ...
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0answers
73 views

Cartan subalgebras of a loop algebra.

For an algebraically closed field $\mathbb F$ of characteristic zero, a finite-dimensional Lie algebra $\frak G$ has a Cartan subalgebra and these subalgebras are conjugated in a certain sense. Let ...
4
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1answer
490 views

Various actions of the Weyl group

Using the notation of my previous question, let $N(T)$ denote the normalizer of the maximal torus $T$ and hence the Weyl group $W(G,T) = N(T)/T$. Here think of roots $\alpha$ as maps $T \rightarrow ...
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1answer
402 views

Central extensions.

Many times I've seen the term "a Lie algebra has a central extension given by" and I got used to it. However, when a Lie algebra has a central extension? Is it unique in some sense?
5
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1answer
319 views

Definition of a “root” of a Lie Algebra

I am using the notation that $g$ is the Lie algebra of the Lie group $G$ and $T$ is the maximal torus of $G$ and $t$ is the Lie algebra of $T$ (and hence $t$ is the Cartan subalgebra of $g$). A ...
0
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1answer
118 views

Lie Algebra of $SL_n(\mathbb H)$

The Lie algebra of $SL_n(\mathbb C)$ are the matrices where the trace is $0$. But what is the Lie algebra of $SL_n(\mathbb H)$ where $\mathbb H$ is the quaternions?
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1answer
416 views

cartan-killing form in $sl_n\mathbb{C}$

Consider $sl_n\mathbb{C}$ as aLie-algebra, and choose h the CSA formed by diagonal matrixes. I can i demonstrate that the Cartan-Killing form in $sl_n\mathbb{C}$ is $<diag(a_i),diag(b_i)>=2n ...
3
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0answers
125 views

Integral forms of loop algebras.

The question following is about integral forms for semisimple Lie algebras and loop algebras constructed from them. Let $\frak g$ a finite-dimensional Lie algebra over $\mathbb C$ and $L(\frak ...
7
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2answers
236 views

Are there finite-dimensional Lie algebras which are not defined over the integers?

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra and let $R \subset \mathbb{C}$ be a subring. Say that $\mathfrak{g}$ is defined over $R$ if there exists a basis $x_1, ... x_n$ for ...
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1answer
125 views

Finding determinant of matrix lie group

Given $G=\left\{A\in M_2(\mathbb{R})\mid A^\top XA = X\right\}$ where $X = \pmatrix{3&1\\1&1}$ and a Lie algebra $\mathfrak g=\left\{Y\in M_2(\mathbb{R})\mid Y^\top X+XY = 0\right\}$, how ...
0
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1answer
130 views

General form of a Lie Algebra

Given $G=\left\{A\in M_2(\mathbb{R})\mid A^\top XA = X\right\}$. Need to find the basis. Error in question
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1answer
181 views

$[y,x]=x$ in a non-abelian solvable Lie algebra

Proof that there exists non-zero elements $x, y$ in a solvable Lie algebra $g$ such that $[y,x]=x$. I have seen an answer from a lecture notes on google, but I can't find it now. Anyway, can someone ...