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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Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) \...
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241 views

Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
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118 views

Weyl orbits of integral dominant weights and convex polytopes

Let $\xi$ be an integral dominant weight of a root system $\Delta$, and let $\mathcal{O}_{\xi}$ be its orbit under the action of the Weyl group. The elements of the orbit are the vertices of an $n$-...
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Weyl's Dimension Formula

I'm trying to grasp how Weyl's Dimension Formula works, and I'm having a bit of trouble. As an example, I was trying to calculate the dimension of V($\varepsilon_1$) for gl(3). First, I set the ...
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104 views

Weyl group of a non-symmetrizable Generalized Cartan Matrix

Let $A$ be a generalized Cartan matrix on the index set $I$. Define the Weyl group of $A$ as the Coxeter group on the basis $I$ with $m(i,j)=2,3,4,6,\infty$ according to whether $A_{ij} A_{ji}$ is $0,...
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A Isomorphism between the extension group and cohomology group of Lie algebras

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove it....
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130 views

Cartan Subalgebra and regular elements

Let $L$ be semisimple Lie algebra, $x\in L$ semisimple. Prove that if $x$ lies in exactly one Cartan subalgebra, then $x$ is regular.
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Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of $\mathfrak{sp}_{2n}\...
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290 views

Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with $\lambda(H_\alpha)...
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Is it possible to further simplify the product of three exponentials $e^A e^B e^C$ when $[A,C]=kB$ (k is a scalar)

The background is calculation of the little group elements of Poincare group for massless particles. I start with a bunch of exponentials of operators, and the end goal is to crunch them into the ...
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275 views

When is the adjoint representation self-dual?

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...
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finding highest weight of dual of a representation of a semisimple lie algebra

If $V$ is an irreducible representation of a semi simple lie algebra having highest weight $\lambda$ then what will be the highest weight of the corresponding irreducible representation $V^*$ (Dual of ...
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450 views

Fundamental vector fields

I have a question related to fundamental vector fields. For that I first setup the notations and properties etc. Let $G$ be a lie group acting smoothly on the manifold $M$. Let $\mathfrak{g}$ be its ...
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443 views

Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.

I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question. To start with we work with the $\mathbb{Q}$ version of Hamilton'...
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108 views

The Nambu bracket

Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $$\{F,G\}_c = \langle\nabla ...
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internal direct product of lie groups

If $G$ is a (edit: simply connected)Lie group, when does a direct sum decomposition of its Lie algebra (into a direct sum of subalgebras) correspond to a (semi)direct product decomposition of $G$? ...
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Proving $\mathscr L(C_G(H)) \subseteq \mathfrak c_{\mathfrak g}(\mathfrak h) = \{ \mathrm x \in \mathfrak g \mid [\mathrm x, \mathfrak h] = 0\}$

Let $H$ be a closed subgroup of the algebraic group $G$, $C = C_G(H)$. Prove that $\mathfrak{c} = \mathscr{L}(C_G(H)) \subseteq \mathfrak{c}_{\mathfrak{g}}(\mathfrak{h}) = \{ \mathrm{x} \in \mathfrak{...
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Why are root systems presented in this confusing way?

I quote Bjorner and Brenti, "Combinatorics of Coxeter Groups." We begin with a simple geometric lemma. Let $m \geq 3$ be an integer, let $\gamma = \pi/m$, and let $k, k'$ be real numbers ...
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434 views

Elementary proof of the third Lie theorem

I would like to understand a (not well known) proof due to G.M; Tuynman, of the third Lie theorem, which asserts that for any given finite dimensional Lie algebra $\mathcal{G}$ there exists a (simply ...
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What are the units of $U(\mathfrak{sl}_2)$?

Let $U(\mathfrak{sl}_2)$ be the Universal Enveloping Algebra of $\mathfrak{sl_2}$ over a field $K$, i.e. the (non-commutative) algebra generated by three generators $E,F,H$ subject to the commutator ...
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Classification of irreducible (g,K)-modules for other g than sl2

Harish Chandra showed how to associate to an admissible representation $(\pi,V)$ of a real semisimple Lie group $G$ the so-called Harish-Chandra module $V_K$ of $K$-finite vectors in $V$. This is a $(\...
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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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The product of dg Lie algebras

I am trying to understand what are products and coproducts in the category of dg Lie algebras. I am okay with coproducts. For products, however, this Wikipedia article says that given $\mathfrak{g},\...
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Why do we need finite dimension for this diagram to commute?

Let $\frak g$ be a Lie algebra and $\rho$ be a finite dimensional representation of $\frak g$ in a linear space $V$. They define the momentum associated to $\rho$ as the map $\mu: V \otimes V^\ast \to ...
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34 views

Definition for Shimura datum

The following definition for $\textbf{shimura datum}$ is due to wikipedia. Let $S=\mathrm{Res}_\mathbb{R}^\mathbb{C}G_m$ be the Weil restriction of the multiplicative group from complex field $\...
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Moment map in general

Let the Lie group $G$ act on the smooth manifold $X$ with the map $(g,x)\to gx$. In any point $x\in X$, the differential of this map induces a linear map: $$ \mu:T_e G \to T_xX\;, $$ and globally, if ...
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Nonstandard analysis, Lie groups and universal enveloping algebras

The idea of nonstandard analysis is to combine finite quantities with infinitesimals. And, back in the day, Lie algebras were roughly considered the "infinitesimal elements" of Lie groups. Say we want ...
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30 views

Hermitian metric from Killing form

Let $G$ be a semisimple Lie group. Its Killing form is a nondegenerate inner product on the tangent space to $G$ at the identity, and this form can be naturally extended to a metric on the whole of $G$...
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Finding the tangent space at the identity *without* using paths

I know that there are two commonly used definitions of the tangent space at a point of a smooth manifold $M$: one using paths through that point, and the other using the idea of linear maps $C^\infty(...
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49 views

Does this binary operation have a name?

Does the following binary operator on abstract linear maps $A,B:\mathbb{C}^N \rightarrow \mathbb{C}^N$, have a name: $[\{A,B\}]:= AB^{\dagger} - BA^{\dagger}$ clearly, it is real bi-linear, but not ...
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Constructing a coset representative of $SO(n,4)/(SO(n) \times SO(4))$.

In $\mathcal N = 2$ Supergravity the scalar components of Hypermultiplets form a quaternionic Kaehler manifold. Only isometries of this so-called target manifold can be gauged. I am interested in ...
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Under what conditions is the homology of a dg coalgebra a graded coalgebra?

I'm trying to get a feel for some differential graded (dg) structures. Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct $\...
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How do roots act on weights?

In Lie theory it's possible to compute things very explicit using tensor methods. For example, we can use an explicit matrix for each generator $T^a$ and compute the "action" of this generator on an ...
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Is it possible to chose the value of $(\alpha, \alpha)\in \mathbb{F}$ for the root system $A_{1}$?

The question is based on what I tried to solve two exercises in James E. Humphreys "Introduction to Lie Algebras and Representation Theory": chapter 26 exerise 1 and chapter 9 exercise 2. I am looking ...
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Real version of Harish-Chandra-Itzykson-Zuber integral

I'm interested in an integral of the form $$ \int_{O(d)} \exp\left(-\frac{1}{2}\mathrm{trace}(CUAU^T)\right)dU $$ where the integration is with respect to the Haar measure on the orthogonal group, i.e....
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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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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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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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39 views

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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39 views

To what extent are the Jordan-Chevalley and Levi Decompositions compatible.

I know that the Jordan-Chevalley decomposition for real Lie algebras only applies to semisimple Lie algebras, but in general the addititive J-C decomposition says that for ANY operator, we can ...
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When did the notion of the tangent space emerge?

When did the modern conception of/notation for the tangent space to a manifold come into use?
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Coproduct of Lie algebras

Fix a commutative ring $k$ and look at the category of Lie algebras over $k$. How do coproducts in that category look like? Notice that what is usually called the "direct sum" of Lie algebras is not ...
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The bijection between central characters and linkage classes over a semisimple Lie algebra

I have a question about the modules over a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra. For a given $\lambda \in \mathfrak{...
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Does the adjoint action induce a trivial action on Lie algebra cohomology?

Let $k$ be an algebraically closed field and $\Gamma$ a finite group. $\Gamma$ acts on itself via conjugation, and it is true that the induced action on the cohomology algebra $H^{*}(\Gamma,k)$ is ...
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Homologie of Lie algebra with coefficients in tensor product of modules

I'd like to prove that $H_i(\mathfrak g, M\otimes N)=\operatorname{Tor}_i^{U \mathfrak g}(M,N).$ My idea was that $H_i(\mathfrak g, M\otimes N)=\operatorname{Tor}_i^{U \mathfrak g}(k,M\otimes N)$. ...
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Computations of common isometry groups, $O(n)/O(n-1), SO(n)/SO(n-1), U(n)/U(n-1)$, etc?

On wikipedia, some of the common isometry groups are given: $S^{n-1}\cong O(n)/O(n-1)$, $S^{n-1}\cong SO(n)/SO(n-1)$, etc. Is there a reference where some/any of these groups are computed? I'm just ...
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Derivative of exponential map

Somehow I've gotten myself confused trying to take the derivative of the exponential map on $\mathfrak{so(3)}$. For vector $\theta$, $\delta \theta$, and $p \in \mathbb{R}^3$, define $$R(\theta, p) =...
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Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
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375 views

SAGE vs. Mathematica for Lie algebras / groups?

What math software is better for working with Lie algebras and Lie groups, SAGE or Mathematica?
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How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...