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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A Lie Algebra $L$ is reductive iff it is completely reducibile as an $\operatorname{ad}_L(L)$-module

Given a Lie Algebra $L$ we say it is reductive if $\operatorname{Rad}L=Z(L)$. How can we prove that $L$ is reductive iff it is an $\operatorname{ad}_L(L)$-module completely reducibile? Suppose $L$ ...
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Prove that actions commute

I am trying to understand a proof from Kobayashi and Nomizu (foundations of differential geometry, p. 280). Suppose that we have Lie subalgebras $a<b<g$, with $g$ the Lie subalgebra of $SO(n)$ ...
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18 views

Rootspace decomposition of a Lie algebra

$ \DeclareMathOperator{\ad}{ad}$ Let $L$ be a non-zero Lie algebra which is semi simple. Then $L$ contains a toral element and hence a non-trivial toral subalgebra. Let $H$ denote a maximal toral ...
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Lie groups and Lie algebras for matrices

Recently, I stumbled over a few things in very basic Lie group / Lie algebra theory concerning matrix groups. Basically, my question is: Is there a way to canonically understand all the Lie groups ...
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46 views

Preimage of singular points of smooth map between manifolds

Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ where $V$ is a (finite dim, real) vector space (of potentially very large dimension) and $SU(n)$ is the special unitary Lie group, what ...
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38 views

Is there a name for this kind of space?

Assume a Riemannian symmetric space $G/H$ where the decomposition of the Lie algebra of $G$ is $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$. It is a known fact that if $\mathfrak{h}$ is the Lie ...
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27 views

Matrix of Killing form a Lie algebra

Let $L$ be the Lie algebra with basis $B = \{u,v,w\}$, with $[u,v] = w, [v,w] = u, [w,u] = v$. Question : Find the matrix of the Killing form $\kappa$ of $L$ with respect to $B$. I have come across ...
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37 views

Unknown proof Lie Algebra

I have a calculation where I do not know what it actually shows. I think it tells me that for right invariant vector fields, the commutator is again right invariant. Maybe somebody here could help me ...
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Weight spaces of a irreducible representation of $\mathfrak{gl}(n, \mathbb{C})$.

Let $\mathfrak{gl}(n,\mathbb{C})$ be the general linear Lie algebra. Let $\{E_{s,t}\}_{1\leq s,t,\leq n}$ be the standard basis for it. And set its Cartan subalgebra $\mathfrak{h}$ to be ...
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29 views

Dimension of Conjugacy class in $SU(n)$

Consider $D \in SU(n)$ ($n$ a multiple of 4), a diagonal matrix with values $\pm 1$ on the diagonal and with trace 0 (only possible for $n$ a multiple of 4). There are $n \choose n/2$ such matrices. ...
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61 views

Symmetric and antisymmetric powers of SU(2) representations

Recently, I took a course in representation theory at Imperial College, and on the first homework the questions were about certain sneaky relationships when it came to representations of SU(2). ...
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Question 3 chapter 4 in Brian Hall's Lie groups, Lie algebras and their representations.

I am not sure how to solve the following exercise from Hall's textbook: Show that the adjoint representation and the standard representation are equivalent reprensentations of the Lie algebra ...
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64 views

Is there an infinite dimensional Lie group associated to the Lie algebra of all vector fields on a manifold?

Since the space $\Gamma(TM)$ of all vector fields on a smooth manifold $M$ is a real Lie algebra with respect to the usual commutator bracket, I was curious if in fact it is the Lie algebra of some ...
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16 views

Duality between the highest weight vector and lowest weight vector.

Let us consider a self conjugate unitary irreducible representation $D$ of a semisimple Lie group $G$ (though I'd be glad if there is a more general case). If $u$ is the highest weight vector of $D$, ...
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21 views

Structure constants for and the adjoint representation and meaning in $sl(2,F)$

First, what I know is that given the basis: $$e = \left(\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right),f = \left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}\right),h = ...
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47 views

An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.

Let $\mathfrak{g}$ be a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. Let $$ 0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 $$ be a short exact sequence of ...
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Flow of a left invariant vector field on a Lie group equipped with left-invariant metric and the group's geodesics

I think the answer to my question is known to many other people, but I'm still getting confused. Let $G$ be a (possibly infinite dimensional also) Lie group and $g$ be its Lie algebra. Consider the ...
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Certain Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$

Is there a lie algebra structure $ [ \;. ] $ on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(\mathbb{S}^{2})$ which is not isomorphic to the standard structures but satisfies the following: ...
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26 views

Globally defined exponential in a particular homogeneous space

I'm currently working in a particular conformal compactification/completion of the Minkowski space-time, but I'm stuck at showing that the exponential of every vector field in it is globally defined. ...
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Why do we need the Dynkin Basis to compute Branching Rules?

Given a representation $R$ of some Lie algbra $g$, we can compute the corresponding representation $R'$ (in general reducible) for some subgroup Lie algebra $ g \supset g'$ by utilizing the weights in ...
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17 views

How to construct generators and Lie Algebra for Lorentz group?

I'm trying to figure out Lorentz group in 2+1. First of all, I am physicist and I'd like to think the special orthgonal group as a combination of rotation and translation in space. Then I construct it ...
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De Rham cohomology of $T^n$ using Künneth formula and Chevalley-Eilenberg theorem.

I want to calculate $H^*(T^n)$ with ring structure using both of these methods. Künneth formula gives $$ H^p(T^n)=H^p(S^1\times T^{n-1})=\bigoplus_{i+j=p}H^i(S^1)\otimes H^j(T^{n-1}) $$ for each ...
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Different Basis/Choices for $SU(3)$ generators?

Conventionally, the generators of $SU(3)$ in the fundamental representation are written in terms of the Gell-Mann matrices. Wikipedia calls this a "particular choice of this representation". What do ...
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39 views

Centre of the derivation Lie algebra

I'm reviewing a paper about Lie algebras for class and I'm finding the following sentence hard to grasp: "It is known and easy to see that if $L = L'$, then $Z(Der(L)) = 0$." where $\mathrm{Der}(L)$ ...
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48 views

commutation relation of angular momentum operator in non cartesian coordinates

The angular momentum operator $J$ in quantum mechanics with the commutation relation \begin{equation*} [J_l,J_m]=i\hbar\epsilon_{lmn}J_n \end{equation*} has the structure of a Lie-algebra. It is ...
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19 views

Why does a simple coroot $\alpha^{(i)}$ correspond to a Cartan subalgebra element $H^i$?

I read here that a simple coroot $\alpha^{(i)}$ corresponds to a Cartan subalgebra element $H^i$ and don't understand why this should be the case. Roots are the weights of the adjoint ...
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Normalizers of subgroups of simple tensors

Inside $GL(2^n,\mathbb{C})$, we have subgroups that are formed by simple tensors. For example, in $GL(16,\mathbb{C})$, we've got subgroups like $H \leq G \leq GL(16,\mathbb{C})$ where $H = \{A_1 ...
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59 views

Derivative of exponential maps in Lie group $G$ and the adjoint operator on its Lie algebra

Let $G$ be a (not necessarily compact, probably even infinite dimensional) Lie group, and $g$ be its Lie algebra. Let $V,W\in g$. Consider $J(t):=(Dexp)_{tV}(tW)$ be the result of differential of the ...
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49 views

Weights in the Dynkin Basis and Eigenvalues of the Cartan Generators for SU(3)?

The Cartan Generators of $SU(3)$ in the three dimensional rep have eigenvalues $(1,-1,0)$ and $\frac{1}{\sqrt{3}} (1,1,-2)$. Therefore we have the weights: $$ (1,\frac{1}{\sqrt{3}}) \quad ...
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40 views

Connected matrix Lie group

While enjoying Lie groups with Brian C. Hall's "Lie groups, Lie algebras, and representations", I'm stuck with the "standard argument using the compactness of the interval $[0,1]$" in the proof of the ...
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1answer
22 views

What really is a path-ordered exponential?

In some texts about gauge theories in Physics I've found one object called a path-ordered exponential which I'm not sure what it means. As I understood, the idea is as follows: let $G$ be a Lie group ...
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scalar multiple of Young symmetriser

The following is a lemma on Fulton and Harris' book -Representation theory,a first course (page 53): Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar ...
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36 views

High Dimensional Rotation Matrices As Product of In-Plane Rotations

Lately I've been thinking a lot about how to find high-dimensional rotation matrices. In particular, can any rotation in $n$-dimensional space be represented as the product of $2$D plane rotations? ...
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$n_-$ is freely generated in a Kac moody Algebra

This question is my doubt from Kac's book on Infinite dimensional Lie algebras. We start with an arbitrary matrix A, and we define the realization of A and using the generators $\{e_i,f_i : 1 \le i ...
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92 views

Right invariance of Casimir (Laplacian)

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. The Casimir element $\Delta\in \mathfrak{zu}(\mathfrak{g})$, considered as an operator on $C^{\infty}(G)$ is right invariant, that is, $\Delta ...
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Recipe to compute dimension and decompose product of $SO(N)$ group representations

As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts: There is a correspondence between irreps ...
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58 views

At what position do we insert the new coefficient in the weights for extended Dynkin Diagrams?

Given a set of weights of a representation and the corresponding extended Dynkin diagram for some Lie algebra, we can delete a node, which yields the maximal subalgebra. I know how to draw the ...
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Adjoint representation and tangent vectors

Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra, $\text{Ad}:G\rightarrow GL(\mathfrak{g})$ the adjoint representation of $G$. Then, for $X,Y\in \mathfrak{g}$, \begin{align*} ...
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32 views

Finite dimensional representations of semi-simple Lie algebras

I've been trying to understand the proof of the following statement: An injective map of $\mathfrak{g}$-representations of a semisimple Lie algebra splits. I'm supposed to show this considering the ...
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42 views

Why can we write the weights of a representation in terms of the simple roots?

I'm currently trying to get my head around the fact that we can write the weights of any representation in terms of the simple roots of the algebra. Is there any, not too-technical, explanation? I ...
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21 views

Complex irreducible representation of solvable lie algebra

How can one infer from the Lie's theorem (in terms of existence of a common eigenvector) that a complex irreducible representation of a solvable lie algebra has dimension 1? What I know is that one ...
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1answer
29 views

L-module definition

I have the following definition of an L-module We say that V is an L-module if there is a k-bilinear mapping L × V → V sending a pair (x, v) ∈ L × V to x.v ∈ V such that [x, y].v = x.(y.v) − y.(x.v) ...
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Perfect Lie algebras

How can I prove that $gl(n,k)$ and $sl(n,k)$ with $[x,y]=xy-yx$ are perfect algebras? By definition ,$g$ is a perfect algebra if $g=g\prime$, where $g\prime=<\{[x,y]| x,y\in g\}.$
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How to prove the Lie bracket is infinitesimal commutator

I am currently studying Lie groups and I cannot solve the following exercise, which I think is vital to my understanding. The Lie bracket is defined as $[X,Y]=\text{ad}(X)Y$. Let the group commutator ...
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23 views

What are some good invariants for low dimensional Lie algebras?

I'm working out some computations on Lie algebras $L$ of low dimensions (by which I mean $3, 4$ or $5$). For my purposes, it is convenient to choose an orthonormal basis $\{e_1, e_2, \ldots, e_n\}$, ...
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43 views

Killing forms and Hermitian inner products

Let $K$ be a compact, connected, simply connected Lie group with Lie algebra $\mathfrak k$ and Killing from $B_{\mathfrak k}$. It is well known that $B_{\mathfrak k}$ is a negative definite symmetric ...
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Subspaces of Lie algebras

The Lie correspondence is well understood. For 'nice enough' Lie groups $G$ (with Lie algebra $\mathfrak{g}$) every sub-group $H < G$ has a Lie algebra $\mathfrak{h} < \mathfrak{g}$ given by ...
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155 views

Linear Algebra : Invertible Matrix Proof

I was doing some linear algebra exercises and came across the following tough problem : Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose ...
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Which elements of $su(n)$ commute with those of a subalgebra $su(2)$

Given a subalgebra $su(2) \subset su(n)$ , how many generators of $su(n)$ commute with any element in the subalgebra $su(2)$? I know that there are at least $n-2$ elements in $su(n)$ satisfying this ...
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51 views

How can I maintain linear independence through a commutator?

Consider a Lie algebra $\mathcal{L}$, a linearly independent generating set $\mathcal{G}$, and an element $X \in \mathcal{L}$. Edit: Note that $\mathcal{G}$ is not necessarily a basis; the generation ...