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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representation theory of two-step nilpotent Lie algebras

Does anyone know of any good reference about the representation theory of two-step nilpotent Lie algebras, like whether their irreducible representations can be classified?
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Every semisimple Lie algebra of dimension at most 5 is simple.

How does one argue that every semisimple Lie algebra of dimension $\leq 5$ is simple. Since any simple algebra has dimension at least $3$, we have to show that any semisimple algebra of dimension $3,4,...
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Matrix representations of $\mathfrak{g}^*$.

Let $g = sl_2$. Then there is a matrix representation of $g$ as follows. The Lie algebra $g$ is a three dimensional vector space with a basis $E, F, H$ such that $[E,F]=H$,$[H,E]=2E$,$[H,F]=-2F$. The ...
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151 views

Sum of nilpotent ideals in a Lie algebra is nilpotent

When trying to show that $I+J$ is nilpotent, whenever $I,J$ are nilpotent ideals of a Lie algebra $L$, I did it brute force: By induction we can show the following: An element of $(I+J)^{2N}$ is a sum ...
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How to understand Weyl chambers? [duplicate]

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane ...
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139 views

Humphreys Introduction to Lie Algebras - Conjugate Borel subalgebras sl(2,F)

Let $L$ be a Lie Algebra and let $E(L)$ denote the subgroup of the inner automorphisms, generated by all $\exp(\operatorname{ad}(z))$ for $z\in L$ being strongly ad-nilpotent. Let $\operatorname{char}\...
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40 views

Weightspace decomposition of a semisimple Lie algebra

$\DeclareMathOperator{\ad}{ad}$ Let $L$ be a (finite dimensional) semisimple Lie algebra. Let $H$ be a maximal toral subalgebra of $L$. Consider a representation $\pi: L \to \mathfrak{gl}(V)$. It is ...
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48 views

dimension of lie algebra

I am studying lie algebra myself and question is about finding dimension of lie algebra . While i read Wikipedia link about lie algebra and lie group i saw statement Lie algebra $\mathfrak{g}$ is ...
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164 views

Fundamental group of a Root System and determinant of the Cartan matrix

This is the first time I am posting, so I hope I didn´t get the formatting wrong. I am currently reading J. E. Humphreys "Introduction to Lie Algebras and Representation Theory" and got stuck at ...
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220 views

Solvable equivalent to nilpotency of first derived Lie algebra?

The Wikipedia "Solvable Lie Algebra" page lists the following property as a notion equivalent to solvability: $\mathfrak{g}$ is solvable iff the first derived algebra $[\mathfrak{g},\,\mathfrak{g}]$ ...
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Etingof problem 2.16.2: Irreps of Two-dimensional Lie algebra over a field of positive characteristic

This is problem 2.16.2 in Etingof's introduction to representation theory. Note that problem 2.16.1 is a proof of Lie's theorem. I'm having trouble with the second case, where the base field has ...
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Description of free Lie algebra in Weibel's book

In Exercise 7.3.2 in Weibel's book An Introduction to Homological algebra the following description of the free Lie algebra over some $k$-module $M$ is given (where $k$ is any commutative ring): ...
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172 views

How to understand Weyl chambers?

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane ...
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1answer
53 views

Possible angles between roots in a root system

Given a Root System $\Phi$ let $\alpha,\beta \in \Phi$ with $\alpha \neq \pm \beta$ and $||\beta||\geq ||\alpha||$. Let $\theta$ be the angle between $\alpha$ and $\beta$. Since $<\alpha,\beta> =...
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$\operatorname{Rad}(k)=\operatorname{Rad}(L)$

Given a Lie Algebra $L$ on a field $F$, we define the radical of $L$ $\operatorname{Rad}(L)$ as the largest solvable ideal of $L$. We define the adjoint representation $\operatorname{ad}_L:L\to\...
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How can Clebsch-Gordan Decompositions be combined?

In section 4 of this paper the authors use a given list of Clebsch-Gordan coefficents for the $27 \otimes 27$ of $E_6$ from an old paper and combine it with their own list of Clebsch-Gordan ...
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59 views

What is corresponding Lie group for Lie algebra of vector fields in dynamical systems?

According to Ado's theorem, for every finite dimensional abstract Lie algebra there is a Lie group. Finite dimensional analytic (or meromorphic) Vector fields (in dynamical systems) over the filed of ...
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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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Question about the theorem of highest weights

I have some confusion from reading Theorem 7.3 in Sepanski's Compact Lie groups and would appreciate it if someone could clarify. In part (e) the book says "for $w\in W$, $wV_\lambda=V_{w\lambda}$, ...
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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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24 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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113 views

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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185 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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42 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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83 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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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 $\bigoplus_{i=...
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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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186 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 $so(3)$...
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103 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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152 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 = \left(\...
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84 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 $\mathfrak{g}$-...
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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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34 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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85 views

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 $p$,...
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65 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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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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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 representation:...
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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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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 (-1,\frac{...
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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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277 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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86 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? I'...
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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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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 ...