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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Problem on Root Systems

Let $R$ be a root system on a Euclidean space $\mathbb{E}$ with simple roots $\{\alpha_i : i=1,....,l\}$. If $\alpha$ = $\sum_{i=1}^{l}c_i\alpha_i$ be a root , then show that $\frac {c_i(\alpha_i|\...
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68 views

How many three and four dimensional Lie algebras are there?

Patera and Winternitz have carried out extensive classification of three and four dimensional Lie algebras. When I tried to look for classification for three dimensional Lie algebra with non-zero ...
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32 views

if $g$ is a lie algebra what is $g^*$?

Iam trying to learn what a coadjoint orbit is but I can't since everywhere I look the definition involves $g^*$.Something that I googled and didn't find anything. I am not even sure what $g^*$. Is it ...
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40 views

Left and right invariant vector fields

I'm following Woodhouse's book on geometric quantization and I'm stuck with this problem. Let $R_A$ and $L_A$ be right and left invariant vector fields such that $R_A(e)=A=L_A(e)$, where $A$ is an ...
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44 views

Correspondence between $\mathfrak{sl}(n,\mathbb{C})$ and $\mathfrak{gl}(n,\mathbb{C})$ representations

There is a one-to-one correspondence between irreducible $\mathfrak{gl}(n,\mathbb{C})-$representations and $\mathbb{C} \times \{ \text{irreducible } \mathfrak{sl}(n,\mathbb{C}) \text{ representations}...
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42 views

extension of derivation of algebras

I am studying extension of derivations, but I am confused of some notations and maybe symbols! For some more details, I recall a theorem. We have the following theorem : Theorem: Let $A$ be an ...
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24 views

Does equivariant property commute with matrix exponential?

Given a vector space $V$ and endomorphism $M : V \to V$ we can define the exponential of $M$: $\exp(M) = \sum_{k=0}^{\infty} \frac{1}{k!}M^k$. This has the property that it commutes with conjugation ...
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Laplacian in matrix form?

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
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Does a subalgebra of a Lie algebra $g$ define a Lie subalgebra in the dual $g^*$ if $( g, g^*)$ is a Lie bialgebra?

Question: Let $\mathfrak d = \mathfrak g\bowtie \mathfrak g^*$ be the double of the Lie bialgebra $(\mathfrak g, \mathfrak g^*)$, and let $\mathfrak h$ be a Lie subalgebra of $\mathfrak g$. If $\xi,\...
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For compact Lie groups, is $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$ equivalent to being semisimple?

In Ana Cannas da Silva’s Lectures on Symplectic Geomertry, page 167, semisimplicity is defined in the restricted setting of compact Lie groups by A compact Lie group $G$ is semisimple if $[\...
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Computing $\operatorname{ad}_x$ where $\operatorname{ad}$ is the adjoint representation [duplicate]

Let $\operatorname{ad}_x:L\rightarrow GL(L)$ be the adjoint representation. In Humphreys "Introduction the Lie-Algebras and representation theory" one can find this example where $x,y$ and $h$ are ...
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30 views

References for the representation theory of $SU(2, 1)$

I couldn't find any reference with the representation theory of this specific case. I found some general stuff but never explicit computations or realizations. The only thing I found on $SU(2, 1)$ ...
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41 views

Taylor Series in any vector space?

I am working through Alexander Kirillov, Jr.'s An Introduction to Lie Groups and Lie Algebras, and on page 29 he does something I find puzzling. He claims that, since the exponential map is a local ...
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1answer
17 views

Every compact semi-simple Lie group has finite center

I am reading introductory lecture notes on Lie groups and Lie algebras. There it is stated as a fact without proof, that any compact semi-simple Lie group has finite center. Here, semi-simple means, ...
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21 views

Difference between symmetry algebra and symmetry group

What is the difference between symmetry algebra and a symmetry group? I just wanted to know if my understanding is right. Lets say we have a system of differential equations. Then the symmetry group ...
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33 views

reference request: lie algebra-lie group

I am looking for a reference where I can find a (relatively) elementary and self contained proof of the fact that all real, finite dimensional Lie algebras are the Lie algebra of some Lie group. ...
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22 views

Adjoint orbit of universal covering group

Let ${\frak g}$ be a complex semisimple Lie algebra and $G$ a connected Lie group with Lie algebra ${\frak g}$. Let $\tilde{G}$ be a universal covering group of $G$. Take $X\in{\frak g}$ and consider ...
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51 views

Weyl's theorem confusion

Weyl's theorem states that given a semisimple Lie algebra $\mathfrak{g}$, any $\mathfrak{g}$-module $V$ is completely reducible. If we consider the case of $\mathfrak{g}= \mathfrak{gl}(1)$, then ...
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Does solvability of Lie algebra have useful application in study of PDEs?

If certain Lie algebra is solvable then what difference this algebra would create in application point view for PDEs ? For example, in case of ODE of fourth order admitting three dimensional solvable ...
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34 views

Why they are Zariski open subsets?

The following is the definition of Zariski topology I am reading a theorem of Lie algebra.In its proof,he says U and R are Zariski open subsets: I have problems in geting the polynomials and ...
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16 views

Irreducible root system decomposition

I am looking for the name of and a good reference on the following theorem Theorem: let $G$ be a connected, compact and semisimple Lie group, and $T \subset G$ a maximal torus of $G$, there exists a ...
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62 views

Lie bracket of $\mathfrak{so}(3)$

I know that for $\mathfrak{so}(3)=\mathcal{L}(SO(3))$, the set of $3\times 3$ real antisymmetric matrices, we can define a basis $$T^1=\begin{pmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{...
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113 views

Anti-involution on universal enveloping algebra of a Lie algebra.

Let $\mathfrak{g}$ finite dimentional semisimple Lie algebra and $\sigma$ the usual chevalley anti-involution that fixes the Cartan subalgebra $\mathfrak{h}$ sends the weight space $\mathfrak{g}_\...
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30 views

How to find a basis of weight vectors

I have to following Lie Algebra $L=\{x\in End(\mathbb{C}^6)\colon x^tS+Sx=0\}$, where $S=[\begin{smallmatrix} 0&I_3 \\ I_3&0 \end{smallmatrix}]$, and the subalgebra $H$ given by the diagonal ...
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Spherical Harmonics and $L_+$ and $L_-$ operators

I have the spherical harmonics $Y_{m}^{l}\left(\theta,\varphi\right)$ and I want to show that the operators $L^{\pm}$ act as "creation" and "annihilation" operators such that $$L^{\pm}Y_{m}^{l}=\sqrt{...
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Determining the compact roots of the Cartan subalgebras of $\mathfrak sp(2,\mathbb R)$

I want to understand the notions of real vs imaginary roots and compact vs noncompact roots (among the imaginary ones) in the theory of Cartan subalgebras (CSA's) of real semisimple Lie algebras. I ...
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136 views

Weyl group, bilinear form, and character/cocharacter pairing. Many questions!

Let $G$ be a connected linear algebraic group, $T$ a maximal torus of $G$, and $\alpha$ a weight of $T$ such that $G_{\alpha} = Z_G(S)$ is not solvable, where $S = (\textrm{Ker } \alpha)^0$. I have ...
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23 views

Decomposition of semi-simple Lie algebras

Background: Let $G$ be a finite-dimensional Lie group, with Lie algebra $L(G)$. A subspace $I\subset L(G)$ which satisfies $[L(G),I]\subset I$ is called an ideal of $L(G)$. A non-abelian Lie algebra ...
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24 views

Reduction of a representation of the Symmetric Group $S_3$

I have this representation of $S_3$ obtained in the usual way $$\varrho\left(\sigma\right)e_i=e_{\sigma_i}$$. Being more explicit the representation is this one: $$\varrho\left(e\right)=\left(\begin{...
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32 views

Killing form on $\mathfrak{sp}(2n)$

I have the same question as this one from a long time ago. Is there an easy way to see that the Killing form on $\mathfrak{sp}(2n)$ is $\kappa(x,y) = (4n+2) \mathrm{tr}(xy)$? For example, the Killing ...
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25 views

Nilpotent Lie Algebras and 2-dimensional Lie Subalgebras

Let be $\mathcal{L}$ a finite-dimensional Lie algebra. How I can prove that if every $2-$dimensional Lie subalgebra of $\mathcal{L}$ is abelian, then $\mathcal{L}$ is nilpotent?
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How to prove that the killing form is unique up to scalar multiple? [duplicate]

For complex simple lie algebra, how to prove that the killing form is the unique adjoint invariant bilinear form up to a scalar multiple. I know we have to use schur's lemma somewhere but don't see ...
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20 views

augmentation ideal of restricted universal enveloping algebra

For restricted Lie algebra $L$ we denote its restricted universal enveloping algebra with $u(L)$. How can we prove that the augmentation ideal has codimension $1$?
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56 views

Is $SO(n)$ actuallly the same as $O(n)$?

$SO(n)$ is defined to be a subgroup of $O(n)$ whose determinant is equal to 1. In fact, the orthogonality of the elements of $O(n)$ demands that all of its members to have determinant of either $1$ or ...
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37 views

Question about calculating Lie bracket given a three dimensional Lie algebra [closed]

Suppose we have $\frak{g}\in\mathbb{R^3}$ spanned by $X, Y, Z$ such that $[X,Y]=Y, [X,Z]=Y+Z$. What is $[Y, Z]$? I tried to expand the bracket, $[X, Y]=XY-YX=Y, [Y, X]=YX-XY$, but don't see how to ...
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83 views

Prove that two Lie groups have homeomorphic universal covers if and only if their corresponding Lie algebra are isomorphic

Two Lie groups $G_1, G_2$ have homeomorphic universal covers $\tilde{G_1}, \tilde{G_2}$ respectively if and only if the corresponding Lie algebras $\frak{g_1}, \frak{g_2}$ are isomorphic as Lie groups....
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Exterior derivative on principal bundle

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
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55 views

Why is Lie algebra a real vector space?

Let the set $\mathcal{g}$ be the Lie algebra of a matrix Lie group $G$. Then my book asserts that $\mathcal{g}$ is a real vector space because it's closed under real scalar multiplication. My question ...
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Why is $T_e \overline{\chi(G)} = \textrm{Im } d \chi$?

Let $G =\textrm{GL}_n$, $s \in G$ diagonalizable, $\sigma: G \rightarrow G$ the automorphism $x \mapsto sxs^{-1}$, and $\chi: G \rightarrow G$ the morphism of varieties $x \mapsto sxs^{-1}x^{-1} = (\...
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35 views

Ideal of a Lie Algebra

I was given this, I think unusual, definition of ideal of a Lie algebra: a subset $I$ of a Lie algebra $L$ is called an ideal if $[I,L]\subseteq I$. I was told from this follows that $I$ is a ...
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33 views

Nilpotent Lie subalgebra of Lie algebra of Killing vector fields

Suppose $M$ is a smooth manifold with Riemannian metric $g$. Recently I have dealt with some problem which lead me to the following question: Can a Lie algebra of Killing vector fields on $M$ has a ...
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35 views

Drinfeld Double definition

A while ago I was doing a reading course on Link Invariants and I came across the notion of a Drinfeld Double: given a Hopf algebra, H, the Drinfeld Double, D(H), was a quasi-triangular Hopf algebra. ...
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Lagrangian densities, Lie Groups and Lie Algebras

I'm quite new to Physics and I was having a look for the first time to the Standard model. I'm not sure if the mechanism that I'm describing is directly from Weyl or from others but what I found quite ...
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76 views

Exponential map for the Lie group of upper triangular matrices

Let $G$ be the Lie group of all upper triangular real matrices (over $\mathbb{R}$) with positive diagonal elements. Denote $\mathfrak{g}$ its Lie algebra. Do we have surjectivity of $\exp : \...
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Simple questions about Cartan subalgebras and root systems

I would ask three questions, but none of these questions are meant to be particular difficult to solve so I figured it would be a waste of space to post three separate threads. (1) Suppose I have a ...
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50 views

Killing form is negative definite

Let $G$ be a compact connected semisimple Lie group and $\frak g$ its Lie algebra. It is known that the Killing form of $\frak g$ is negative definite. What about the Killing form $B$ of the complex ...
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1answer
23 views

Showing a set is a root system in a vector space from definition of root system

Suppose I have the vectors $\alpha, \beta \in \mathbb{R}^2$ with inner products $(\alpha, \alpha) = 1$ and $(\beta, \beta) = 2$, and the angle between $\alpha$ and $\beta$ is $\theta = \frac{3\pi}{4}$....
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1answer
20 views

Simplicity of $\mathfrak{so}(1,3)$

I am trying to solve an exercise asking to determine if $\mathfrak{so}(1,3)$ is simple or semisimple as real Lie algebra but I am having troubles. My idea is to prove $\mathfrak{so}(1,3)$ is simple ...
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28 views

Decomposing representations

The problem I am trying to do is the following: Show that vector representation 5 and adjoint representation 10 in SO(5) decompose respectively into representations of SO(4) as: 5 →4⊕1 10→6⊕4 I ...
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10 views

derivation of restricted Lie algebras

Let $L$ be a restricted Lie algebra and $A$ be a subalgebra of $L$ What is the description for $\delta$ as a derivation of the $p$-algebra $A$ into $L$? In other words, according to the definition of ...