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Questions tagged [lie-algebras]

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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For $G$ a real connected solvable Lie group, the commutator group $[G,G]$ is nilpotent.

I was stuck on showing the following problem: For $G$ a real connected solvable Lie group, the commutator group $[G,G]$ is nilpotent. There are a few ways I thought about this problem: Approach ...
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1answer
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Left-/right-translate of a two-form

The context is that of coboundary Lie bialgebras discussed in "Lie bialgebras, Poisson Lie groups and dressing transformations" by Y. Kosmann-Schwarzbach. In section 4.2, she defines objects like $r^...
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Proof of an identity for the Killing form involving derivations.

I'm working through Ziller's Lie Groups. Representation Theory and Symmetric Spaces, and in Proposition 1.36, he shows the following identity: Let $\mathfrak{g}$ be a real or complex [finite-...
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Applications of the PBW theorem

What are some nice corollaries or applications of the PBW theorem? There's this immediate corollary that a Lie algebra sits inside the universal enveloping algebra so in particular, the Lie algebra ...
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1answer
24 views

Quotient of a quotient.

Let $\mathfrak{g}$ be a complex reductive Lie algebra. Let $M,N,L$ be finite dimensional $\mathfrak{g}$-modules. Suppose $M$ is a quotient of $N$ and $N$ is a quotient of $L$. My question: Is $M$ a ...
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19 views

Is this subalgebra semisimple?

Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Let $\Phi$ be the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by ...
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Angle-Axis Parametrization of SO(3) Proof

Suppose we have an element $R$ of $SO(3)$. $R$ is characterized by, $R^T = R^{-1}$. There are a number of equivalent characterizations such as $R$ preserves norms or dot products. I am looking for a ...
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Composition of uncertain rotations

Here, in this tutorial: http://ethaneade.com/lie.pdf, the author gives the composition of uncertain rotations for Gaussians in $SO(3)$ (Eqn. 47, Page 7). The author doesn't give the detailed ...
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Dual Space to a Lie Algebra. Problem from Takhtajan. Finding the center of a Poisson Algebra

So I am trying to solve Problem 2.19 from the book "Quantum mechanics for mathematicians" by Takhtajan. The problem is the following: Let $g$ be a finite-dimensional Lie Algebra with a Lie bracket $[,...
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1answer
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Doubt in understading Proof of Matrix lie group and Lie algebra locally homemorphic

I was reading Brian C Hall Lie Group book In that I encountered following proof . I understand Whole proof. But have one doubt Why Auther take Orthogonal complement into consideration As I think ...
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Every Borel contains a Cartan, and conjugacy theorems: A simple proof?

Conjugacy of Borel subalgebras $\newcommand{\ad}{\mathrm{ad}\,}$ Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. A Borel subalgebra $\mathfrak{b} \subseteq \mathfrak{g}$ is a ...
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$\mathbb R$-points of semisimple real algebraic groups, connectivity, and Cartan involutions: some questions

I am reading about Cartan involutions on semisimple real Lie groups and have a point of confusion I am trying to reconcile with linear algebraic groups. Let $\mathbf G$ be a linear algebraic group ...
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Deciding whether a representation is orthogonal or symplectic

I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement ...
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1answer
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An operator exponential/commutator question

There is "an important lemma" related to the Baker-Campbell-Haussdorff theorem which says that $$ e^XYe^{-X} = Y + [X,Y] + \frac{1}{2!}[X,[X,Y]]+\ldots $$ Clearly if $[X,Y]=0$ we get (noting that $e^...
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Why is Jacobi Identity equivalent to holonomy of system? [on hold]

Or equivalently, why is jacobi identity equivalent to integrability of system? How do I understand it intuitively? Thanks.
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What is the group name of matrixex that are both hermitian and unitary [closed]

I am just curious about the homotopy group of the matrices that are both hermitian and unitary. To me it seems that it is not just the subgroup of U(N) with either determinant of 1 or -1. So what is ...
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For which graphs does this “+1 game” terminate?

Consider this game on simple graphs described by Allen Knutson: Begin by assigning a $1$ to a single node and a $0$ to each other node in the graph. Then, while such a node exists, choose a node with ...
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39 views

Simple formula for the dimension of weight spaces of Verma module?

Let $\mathfrak{g}$ be a simple Lie algebra (e.g. $\mathfrak{sl}_n$), and let $M_\lambda$ be the Verma module with highest weight $\lambda$. Is there a simple formula for $\dim (M_\lambda)_\mu$, where ...
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1answer
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About Weyl group [closed]

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Let $W$ be its Weyl group. I would like to know whether $W$ is always finite? If so, why?
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Examples of three dimensional Non Nilpotent Leibniz or Lie algebras.

Can one give me some examples of three dimensional Non Nilpotent Leibniz algebras? Any references to the classification of three dimensional Non Nilpotent Leibniz or Lie algebras will also be helpful.
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3answers
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Vector Field Exponential Map

I've got ${\bf v} = x^2\partial_x$, and I'm trying to find $\exp(\varepsilon{\bf v})$, but I'm having some trouble. If I define ${\bf v}^{n+1} = {\bf v}{\bf v}^n$ then I get a different outcome to ${\...
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Definition of Lie Algebroid, pushforward of smooth vector fields is vector field?

This is the main question: if $p:A \rightarrow B$ is a smooth vector bundle homomoprhism over base space $M$, then $pX$ is a smooth section of $B$, where $X \in \Gamma(A)$ is a smooth section of $...
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1answer
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Lie bracket of canonical vectors on tangent space to a point on a manifold is zero.

Let M be a manifold and $T_p(M)$ be the tangent space at $p$, and $\phi$ a local chart around $p$. Let $$\left.\frac{\partial}{\partial\phi^1}\right|_{_p},\ \cdots\ ,\left.\frac{\partial}{\partial\...
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Intuition behind adjoint map

Let $G$ be a Lie group, and let $g,h\in G$. Suppose we have the map $$\Lambda_g:G\to G$$ such that $$h\to ghg^{-1}$$ This induces a map $\mathfrak{ad}_g$ on the tangent spaces such that $$\mathfrak{dg}...
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Lie Algebra paper

I recently finished reading Erdmann and Wildon's book on Lie Algebra. I am supposed to present a paper in a student talk. I am looking for papers that are accessible to me. Can someone suggest any ...
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Norm minimization with Lie theory

Consider the following flow on square matrices. $$Y(t)= \frac{d}{dt}X(t)$$ I would like to find a skew-Hermitian matrix $K$ which minimizes the following under the Frobenius norm. $$||[K,X]-Y||_F$$ We ...
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Factorization of Lie group elements

I am reading a paper that does the following: Constructs a Lie algebra $\mathfrak{g}$ of derivations acting on a certain space (as an infinite direct sum of one-dimensional spaces of derivations) ...
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25 views

Proof of Weyl Character formula for $sl_n\mathbb{C}$

Is there a proof of Weyl character formula just for $sl_n\mathbb{C}$ independent of any 'heavy machinery'? Please suggest some references if possible.
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“If $g$ is semisimple, It is not too hard to see that $H^2(g,a)=0$. With a little supplementary argument…”

This is a statement made in Knapp, Lie groups, Lie algebras, Cohomology Chpt 4 last paragraph of Sec 2. $H^i(g,a)$ is the $i-$th cohomology group of complex $Hom(\wedge^i g,a)$ with $a$ abelian lie ...
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Show $H \cap L_i$ is a Cartan subalgebra of $L_i$

Suppose we have a semisimple complex Lie algebra $L$, with a Cartan subalgebra $H$. Suppose that $L= L_1 \oplus\cdots\oplus L_k$ with each $L_i$ a simple ideal of $L$. I want to show that $H_i=H \...
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1answer
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name of Lie group $\text{SU}(2) \times \text{SU}(2) \times \text{SU}(2)/\mathbb{Z}_2$

Once I read a paper in which the author(s) gave a sensible (possibly well-established) name to the $9$-dimensional compact Lie group $$\frac{\text{SU}(2) \times \text{SU}(2) \times \text{SU}(2)}{\...
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1answer
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Conjugacy of Borel Subalgebras: Proof in Humphreys' Introduction to Lie Algebras and Representation Theory

In the title referenced above a proof of the conjugacy of Borel subalgebras is given on page 84: We assume $L$ semisimple and let $B$ be a standard Borel subalgebra and $B'$ any other Borel ...
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Representation of Lie algebra of germs of smooth/holomorphic functions

$\def\O{\mathcal{O}} \def\g{\mathfrak{g}}$ Suppose $G$ is a real or complex Lie group, with Lie algebra $\g$. Write $\O_{G,1}$ (resp. $\O_{\g,0}$) for the ring of germs of smooth/holomorphic function ...
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Question regarding the definition of the action of $sl_2$ on a vector space

Let $L$ be a simple $sl_2$-module of finite dimension over a field $K$ of characteristic $0$. Say $e,f,h$ is a basis of $sl_2$. In the notes I am reading it only defines the action of $e,f,h$ on the ...
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Riemannian metric and Laplacian coming from an invariant form on the lie algebra

Let $G$ be a semisimple real Lie group. Let $\Delta \in U(\mathfrak g_{\mathbb C})$ be the Casimir element associated to the Killing form on the complexified Lie algebra $\mathfrak g_{\mathbb C}$ of $...
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1answer
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Show $\left<h\right>_{\mathbb{C}}$ is a Cartan subalgebra of $\mathfrak{sl}\left( 2,\mathbb{C}\right)$.

Using the standard basis elements $$e=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, f=\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},h=\begin{pmatrix} 1 & 0\\ 0 & -1 \end{...
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1answer
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Question about definition related to root system of semisimple Lie algebras

Let $L$ be a semisimple Lie algebra of finite dimension over a field of charcteristic 0 and algebraically closed, and $H$ a maximal toral subalgebra. Let $R$ be the set of roots of $L$ with respect ...
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1answer
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How to show a vector space is a real form of another?

I only know from the definition that, say $V_1$ is a real form of $V$, if $V=\mathbb{C}\otimes_\mathbb{R} V_1$, but what does this really mean? Is it true it is just saying that $V$ is spanned by ...
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KIlling vectors from isometries and orbit spaces

I am currently (trying) to learn more about orbit spaces generated from an isometry group of a manifold. I cannot quite pinpoint what I (don't) understand, so I will try to lay out what I could gather:...
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How to show this $L$-module is simple? (related to the root space decomposition of semisimle Lie algebras)

Given a semisimple Lie algebra (finite dimensional over a field $K$ characteristic $0$ and algebraically closed), there exists a root space decomposition $$ L = H \oplus \oplus_{\alpha \in R} L_{\...
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1answer
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Proof the Lie algebra morphisms induce algebra morphisms on the universal enveloping algebra.

I am struggling to understand Theorem V.2.1 in Christian Kassel's Quantum Groups page 95. The Theorem is stated as follows. Let $L$ be a Lie algebra. Given any associative algebra $A$ and any ...
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What does the “standard basis” of $O(1,n)$ mean?

Let $O(1,n)$ be the orthogonal group of the quadratic form $b(x)=-x_0^2+\sum\limits_{i=1}^n x_i^2$. In other words, $$O(1,n)=\{T\in GL(n+1,\Bbb{R}|b(Tx)=b(x),\forall x\in R^{n+1}\}$$ Elements in $O(1,...
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Complex conjugated representation and Young tableaux

Imagine you have the Young tableu and the Dynkin numbers, $(q_1, q_2, ..., q_r)$, of the Lie algebra of $SU(n)$ which has $r$ simple roots. The way I assign Dynkin numbers is increasing its value from ...
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1answer
35 views

Examples of Lie algebras of the $BC$ root system type

What are some examples of Lie algebras of the $BC$ root system type please? I am actually interested in the corresponding groups too. I heard that there were Lie algebras over $\mathbb{R}$ having $BC$ ...
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Is the semisimple part of $x \in L$ belong to $L$ when $L$ is a semisimple Lie algebra?

Let $L$ be a finite dimensional semisimple Lie algebra over a field of charcteristic $0$ and algebraically closed. I have learned that the map $$ ad: L \rightarrow \ Der(L) $$ is an isompsphism, ...
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38 views

Relationship of SU(2) & SU(2) to SO(4)?

While almost all accessible references indicate/demonstrate that group SO(4) = SU(2)⊗SU(2), I've come across two references that state the relationship as SO(4) = SU(2)⊕SU(2). Is the latter equation ...
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1answer
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Geometric interpretation of the second Bianchi identity?

Assuming a torsion free Christoffel symbol, the covariant derivative can be shown to satisfy the second (differential) Bianchi identity: \begin{equation} [[\nabla_a,\nabla_b],\nabla_c]+[[\nabla_c,\...
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1answer
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Understanding the proof of the theorem that the exponential map is locally one-to-one at 1 and onto

I am trying to understand a piece of the proof that the exponential map is locally one-to-one and onto in Brian Hall's Lie Groups, Lie Algebras, and Representations. Theorem 3.42. For $\epsilon\in(...
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Group characters are eigenfunctions of the Laplacian with eigenvalue proportional to the quadratic Casimir

For the finite dimensional irreducible representations of $SU(2)$ we have that the group characters $\chi_n(g)$ for the $n^{th}$ representation are eigenfunctions of the Laplacian over the group ...
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If a $k$-vector space $V$ is a simple $\mathfrak{sl}_2$-module then so is $V \otimes \bar{k}$?

Let $V$ be a finite dimensional $k$-vector space which is a simple $\mathfrak{sl}_2$-module. Here $k$ is a field of characteristic $0$ and we let $\bar{k}$ denote its algebraic closure. I was ...