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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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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14 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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How to prove that the killing form is unique up to scalar multiple?

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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49 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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34 views

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

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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59 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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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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49 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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16 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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32 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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27 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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32 views

Dimension of maximal tori

Let $G$ be a compact Lie group. $T$ $-$ its maximal torus. Is there a simple reasoning to show that dimensions of $T$ and $G$ have the same parity? I am sorry if this quesion is for children, but ...
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59 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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39 views

Classification Problems in Lie algebra

My field of interest is Lie group analysis of PDEs, currently I am struggling with techniques for classification of Lie algebra into mutually conjugate classes of 1- and 2-dimensional sub-algebras ...
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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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47 views

Ordering on the weight lattice

When given a finite dimensional complex Lie algebra $\mathfrak{g}$ that is also semisimple and a choice of Cartan subalgebra $\mathfrak{h}$ we may talk about its weight lattice $\Lambda_{W} $ in ...
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Maple: How to use partial differential operators?

I am trying to calculate the commutator $[v,w]=vw-wv$ for given infinitesimals $$v=\dfrac{\partial}{\partial x}$$ and $$w=x\dfrac{\partial}{\partial t}$$ I know how to calculate the commutator by ...
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If $H$ is a diagonal matrix then its adjoint representation is also diagonal for any $H$ [on hold]

I am not sure why this is true, is there a simple explanation? To be more precise I am trying to show that the set of traceless diagonal matrices form a Cartan subalgebra of $sl(n,\mathbb{C})$.
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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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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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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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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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1answer
22 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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18 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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21 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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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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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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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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19 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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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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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 $\overline{ad(d)} = ...
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Adjoint action on Lie algebra su(2) ($A \in SU(2), X \in \mathfrak{su(2)} \Rightarrow AXA^{-1}\in \mathfrak{su(2)}$)

I am trying to understand ho $SU(2)/\{\pm I\} \cong SO(3)$ (see: how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$) but i am not sure about the adjoint action. In especially, as I understand, the adjoint ...
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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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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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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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50 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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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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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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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 ...
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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 ...
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Matrix exponential between Lie algebra and Lie group (help with a proof)

Theorem 3.42 in Hall's Lie Groups, Lie Algebras and Representations is a key step towards proving that the matrix exponential maps a neighbourhood of zero in the Lie algebra to a neighbourhood of the ...
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Expression of the Laplacian of the reduced Heisenberg group?

Let $\mathbb C^n$ be the n-dimensional complex field endowed with a positive definite hermitian form $H(z,w)$. The corresponding symplectic form is $E(z,w)= \Im (H(z,w))$, where $\Im $ denotes the ...
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How $U_{q}(\mathfrak{sl}_{2})$ becomes the universal enveloping algebra $U(\mathfrak{sl}_{2})$ of $\mathfrak{sl}_{2}$

My question is how $U_{q}(\mathfrak{sl}_{2})$ becomes the universal enveloping algebra $U(\mathfrak{sl}_{2})$ of $\mathfrak{sl}_{2}$ if we set $t=q^h$ and $q$ tends to 1.
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Lie Algebra: Optimal system of one-dimensional sub-algebras of the heat equation

This is a follow up question to Invariants of a PDE by Lie Symmetries, as I tried to follow the reasoning from the book Applications of Lie Groups to Differential Equations (Peter J. Olver, Example ...
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Show that $\pi(Z)$ acts as a scalar over $\mathbb{g}$

Let $(\pi, V)$ be a finite dimensional irreducible representation of $\mathbb{g}$ $V$ is a vector space of homogeneous polynomials in 3 variables of degree d over $\mathbb{R}$ ...
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Does the trivial character always show up as a weight?

Let $G$ be a linear algebraic group, $T$ a subtorus of $G$ of dimension $\geq 1$. Let $\mathfrak g$ be the Lie algebra of $G$. Then the Ad operator $$\textrm{Ad } : G \rightarrow ...
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Solving a PDE to Yield Determining Equations

I'm going through an example in Peter Hydon's book "Symmetry Methods for Differential Equations" which finds the basis for the Lie Algebra of the point symmetry generators for Burgers' equations. ...
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Connecting the regular representation of $\mathfrak{so}(3)$ and the exterior algebra of $\mathbb{R}^3$

It is well known that the regular representation of $\mathfrak{so}(3)$ is the so-called "cross product" matrix $A(x)$ which follows $A(x)y = x\times y$, and $x,y\in\mathbb{R}^3$, while the cross ...
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Meaning of generators in Lie Algebra of PDE

Consider some PDE involving a scalar function $u(x,t)$ with two independent variables $x$ and $t$. Assume that this PDE has a Lie Algebra spanned by the following generators, $X_1=\partial_x,\quad ...
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Soft question about Lie Groups and 3D rotation

Let $R(\phi, \boldsymbol{n})$ be a member of Lie Group SO(3). According to Wikipedia If $R(\phi, \boldsymbol{n})$ denotes a counter-clockwise 3D rotation through an angle $\phi$ about the axis ...