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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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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1answer
21 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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1answer
15 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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1answer
31 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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30 views

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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1answer
30 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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15 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 ...
2
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1answer
55 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\\ ...
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1answer
89 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 ...
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23 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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26 views

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 ...
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10 views

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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83 views
+500

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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0answers
21 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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1answer
21 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: ...
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1answer
27 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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1answer
22 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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15 views

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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1answer
16 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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1answer
54 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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1answer
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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1answer
70 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 ...
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49 views

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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1answer
53 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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1answer
18 views

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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1answer
29 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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1answer
27 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. ...
2
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31 views

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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1answer
62 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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19 views

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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2answers
47 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 = ...
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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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1answer
26 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 ...
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13 views

Subrepresentation of invariants in hom space between irreducible representations

Let $\mathfrak{g}_1, \mathfrak{g}_2$ be semisimple lie algebras with irreducible representations $U$ and $W$. Write $\mathfrak{g} = \mathfrak{g}_1 \oplus \mathfrak{g}_2$ and consider both of the ...
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12 views

Casimir operator for direct sum of lie algebras

I'm revising for my exams now and have run into a bit of a problem. If $\mathfrak{g} = \mathfrak{g}_1 \oplus \mathfrak{g}_2$ is a direct sum of semisimple Lie algebras and $(V,\rho)$ is a ...
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25 views

What does it mean for a representation to be one-dimensional?

For example, take the Heisenberg Lie Algebra with the following basis: $X=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ $Y=\begin{bmatrix} 0 ...
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1answer
20 views

Lie algebra homomorphism and representation

I am solving a multiple part problem on Lie algebra representations. I have done the first three parts, but am stuck on part (iv) as follows: Define a linear map $\phi : \mathbb{g} \rightarrow ...
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37 views

Defining a filtration on free Lie algebra in terms of generators

Let $L$ be the free Lie algebra generated by the set $X = \{x_i\}_{i \in \mathbb{N}}$ over a field $k$. A Lie monomial is a bracketed word of elements of $X$ of finite length, and $L$ is spanned by ...
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1answer
34 views

Very basic Lie Algebras question on the complex conjugate of the adjoint map

$V$ vector space over $\mathbb{C}$, $L$ Lie subalgebra (subspace and closed under Lie bracket) of $gl(V)$, linear maps $V \to V$. Suppose $d \in L$ diagonalisable, show that $\overline{ad(d)} = ...
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2answers
78 views

Proving a Jacobi-like identity with four elements in a Lie algebra

Let ${\frak g}$ be a Lie algebra. Then $$ [[[X,Y],Z],W] +[[[Y,X],W],Z]+[[[Z,W],X],Y]+[[[W,Z],Y],X]=0 $$for all $X,Y,Z,W \in {\frak g}$. I started using the Jacobi identity four times to get: ...
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11 views

Lie Algebra: Dimension of a one-parameter group (dimension of orbit)?

I am reading a book about Applications of Lie Groups to Differential Equations by Peter Olver. Lets say we have a PDE with $p$ independent variables and $q$ dependent variables In Chapter 3.1 (page ...
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1answer
51 views

Finding lie algebra of a group by using exp map and tangent space [duplicate]

I'm studying Lie groups and I am in trouble with finding lie algebras of the classical groups. How can I calculate $\mathfrak{sp}(n,\mathbb{C})$ or $\mathfrak{so}(n,\mathbb{R})$ using exp map and ...
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24 views

What does the set of dominant integral elements in a Cartan sub algebra look like?

I'm reading about the theorem of the highest weight: Irreducible finite dimensional representations of a complex semisimple Lie algebra (with a fixed Cartan sub algebra, $\frak{h}$ and choice of ...
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18 views

The smallest $m \in \mathbb{N}$ such that $b(n, \mathbb{C})$ is soluble.

Say I have the Lie algebra $L = b(n, \mathbb{C})$, the set of all $n \times n$ matrices with entries in $\mathbb{C}$ that are upper triangular with the standard Lie bracket (the commutator $[A, B] = ...
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49 views

A soft question on Gauge Equivalence in Integrable Systems

I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations ...
2
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1answer
33 views

Endomorphism vector bundle isomorphic to the adjoint bundle of its frame bundle?

Could somebody help me to prove the following isomorphism (in particular what is the isomorphism)? \begin{equation} End(\xi) \cong ad(E_{\xi}) = E_{\xi} \times_{GL(n,\mathbb{R})} ...
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28 views

Is the BGG category $\mathcal{O}$ a Serre subcategory of $\mathfrak{g}$-mod? [duplicate]

Let $\mathcal{O}$ be the BGG category for a be a finite-dimensional, semi-simple complex Lie algebra $\mathfrak{g}$. Let $\mathfrak{g}$-mod be the category of all $\mathfrak{g}$-modules. Is the BGG ...