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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Algebra of matrix coefficient over a compact group is isomorphic to its dual.

Let $K$ be a compact group. Then we have the following definition of matrix coefficient: Definition: $f: K \rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite dimensional ...
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What are some good books to study finite dimensional lie algebras?

I like conceptual books, which have helpful examples too. Thanks
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1answer
17 views

Symmetrizability of generalised Cartan matrix

How to prove that a generalized Cartan matrix whose diagram contains no cycles is symmetrizable? Any hint would be sufficient. Thanks in Advance.
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How to find coefficients in Lie bracket relations in Cartan-Weyl basis?

For example, consider an $D_n$ Lie algebra. The Cartan-Weyl basis satisfies the following Lie bracket relations [1, p.98] $$ \left[H^{\alpha},H^{\beta}\right]=0 $$ $$ ...
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1answer
42 views

Prove an identity using differential calculus to a problem connected to fluids

Euler's equation for a incompressible inviscid fluid is $\displaystyle \frac{\partial \textbf{v}_t}{\partial t}+(\textbf{v}_t \cdot \nabla)\textbf{v}_t=-\nabla p_t$ where ...
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What is $\Lambda^{i}$ in the “Show that the highest weight of $\Lambda^{i}V $ is $\omega_{i}”$?

Question: What is $\Lambda^{i}$ in the "Show that the highest weight of $\Lambda^{i}V $ is $\omega_{i}$"? In this question, $\omega_{i}$ are fundamental weights. Context: Highest weight modules of ...
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1answer
67 views

How to show trace of $AB$ is zero for $A \in \mathfrak{u}_n$ and $B \in \mathcal{H}_n$?

Please have a look at this question: Help needed in understanding the basics of Cartan decomposition of a Lie algebra I want to show that the decomposition $\mathfrak{gl}_n = u_n \oplus ...
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Help needed in understanding the basics of Cartan decomposition of a Lie algebra

I am trying to learn the basics of Cartan decomposition of Lie algebra, and have come across the following example. Consider $\mathfrak{gl_n}$ as the Lie algebra of endomorphisms of $\mathbb{C}^n$. ...
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26 views

simple lie algebras

Is there any way of directly proving that the lie algebra $\mathfrak{sl}(n,\Bbb C)$ is simple? I am not asking for a complete proof, but could somebody please give me a hint on how I can proceed?
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Dimension of image of Lie bracket

Is there a method to calculate the dimension of the set of vectors in $\mathfrak{su}(n)$ $\{\ [A,B] \ \text{s.t} \ B \in \mathfrak{su}(n)\}$ for some fixed $A$. Is the dimension the same for all $A$?
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Show that $\Phi^*_t\mu=\mu \iff \mathrm{div}_{\mu}X=0$.

Let $\mu$ be a non-vanishing $1$-form on $\mathbb{R}^n$. Given a smooth vector field X on $\mathbb{R}^n$, we define the divergence of $X$ wrt $\mu$, denoted by $\mathrm{div}_{\mu}X$, by $L_X \mu = ...
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Relation between simple roots and fundamental weights.

Let $\alpha_1, \ldots, \alpha_n$ be simple roots of a semisimple complex Lie algebra. Let $\omega_1, \ldots, \omega_n$ be the fundamental weights. We have $$ \alpha_i = \sum_{s} k_s \omega_s, $$ for ...
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Does an arbitrary product of $f$ and $f^\dagger$ belong to a universal enveloping algebra of the Heisenberg algebra?

The Heisenberg algebra is essentially the canonical commutation relations (CCR) for bosons $[f,f^\dagger]=1$. $f$ is called an annihilation operator in physics ($f^\dagger$ creation operator). ...
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Why is this generator an element of the Cartan subalgebra of SU(3)?

In the solution of Problem 3 of these notes (papge 4), it stated that $\lambda_2$, $\lambda_5$ and $\lambda_7$ form a SU(2) subalgebra of SU(8), where $\lambda_i$ are the Gell-Mann matrices. In ...
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Are there common inequivalent definitions of Cartan subalgebra of a real Lie algebra?

I'm confused about seemingly different notions of a Cartan subalgebra of a real semisimple Lie algebra, and I'm wondering if there are common inequivalent definitions. In the book Lie Groups: Beyond ...
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representation $\pi_{m,\,n}: \text{SU}(n) \to \text{GL}(V_m)$

Let $V_{m,\,n}$ denote the vector space of the homogeneous complex polynomials of degree $m$ in $n$ variables (under addition). Define a representation $\pi_{m,\,n}: \text{SU}(n) \to ...
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Character of a tensor product of $\mathfrak{sl}_2$-modules

Let $V$ be a finite-dimensional $\mathfrak{sl}_2$-module. There is a standard base $\{e,f,h\}$ in $\mathfrak{sl}_2$, I use standard notation ($h$, for instance, is the diagonal matrix with $1$ and ...
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1answer
68 views

Proving $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$

Let $X,Y$ be vector fields. $L_X$ is the Lie derivative and $i_X$ is the contraction of a $k$-form. I am really stuck on how you could prove the identity $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$. Update: I ...
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28 views

Proving $L_{\mathbb{X}}i_{\mathbb{Y}}=i_{[\mathbb{X},\mathbb{Y}]}+i_{\mathbb{Y}}L_{\mathbb{X}}$ [duplicate]

Let $\mathbb{X}$, $\mathbb{Y}$ denote vector fields on $U \subset \mathbb{R}^n$. Prove the identity $L_{\mathbb{X}}i_{\mathbb{Y}}=i_{[\mathbb{X},\mathbb{Y}]}+i_{\mathbb{Y}}L_{\mathbb{X}}$ I ...
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Proving that $\Phi_{t*}[\mathbb{Y},\mathbb{Z}]=[\Phi_{t*}\mathbb{Y},\Phi_{t*}\mathbb{Z}]$

Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$. Let $\Phi_t$ denote the flow of $\mathbb{X}$. You are given that $\displaystyle L^j_{\mathbb{X}}[\mathbb{Y},\mathbb{Z}]=\sum_{k=0}^{j} ...
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1answer
23 views

Showing that $F^*(L_{\mathbb{Y}}\omega)=L_{\mathbb{X}}(F^*\omega)$

Let $F:U \rightarrow V$ be a diffeomorphism between open sets in $\mathbb{R}^n$. Let $\mathbb{Y}$ be a vector field on $V$ and $\omega$ a $k$-form on $V$. Show that ...
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1answer
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Why is the commutator subalgebra of a Lie group a linear subspace?

One defines commutator subalgebra of Lie algebra $\mathfrak{g}$ as $[\mathfrak{g},\mathfrak{g}]$. Why is it really subalgebra: Why $\forall_{a,b,c,d \in \mathfrak{g}} \exists_{e,f \in \mathfrak{g}} ...
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An exponential map of a matrix computation

Suppose the $n\times n$ matrices $A$ and $M$ satisfy $AM+MA^{T}=0.$ Show by direct computation that the product $\mathrm{exp}(At)~M~\mathrm{exp}(A^{T}t)=M$ for all $t\in \mathbb{R}.$ Note: By ...
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Why the Steinberg idempotent is idempotent?

Consider the group $GL_n(\mathbb{F}_p)$. We have the following subgroups : -$\Sigma_n$ the symmetric group (permutation matrices) -$B_n$ the Borel subgroup (upper triangular matrices) -$U_n$ the ...
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1answer
32 views

What is an example of a map not satisfying this rank condition?

Definition: Consider a Lie Group $G$ and a set of right invariant vector fields on $G$, denoted $\Gamma$. A point $y \in G$ is called normally accessible from a point $x \in G$ by $\Gamma$ if there ...
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1answer
11 views

Definition of the Dynkin Diagram (in Humphreys)

I'm reading paragraph 11 in Humphreys' 'Introduction to Lie Algebras and Representation Theory'. The author defines Coxeter graphs and Dynkin diagrams for any rank-many distinct positive roots. He ...
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Why are lie algebra of upper-triangular $nxn$ matrices not nilpotent Lie algebra

Is there an easy proof (without Engel's theorem) of the fact that lie algebra of upper-triangular $n\times n$ matrices (of the field $\mathbb{R}$) are not nilpotent Lie algebra?
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Subalgebra condition in Engel's theorem

An equivalent version of Engel's theorem says that Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V\ne 0$, then there exists ...
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Lie subalgebra in $Der(\mathbb{C}[z])$ isomorphic to $\mathfrak{sl}_2$

I am to prove that $\{(az^2+bz+c)\frac{\partial}{\partial z}:a,b,c\in\mathbb{C}\}$ regarded as a Lie algebra is isomorphic to $\mathfrak{sl}_2(\mathbb{C})$. I guess it is possible to build a basis ...
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if X \in gl(V) is any nilpotent element, then the adjoint action ad(X) is nilpotent

I found a observation in the beginning of the proof of Engel's Theorem in the Fulton's book "Representation Theory" Observation: if $X \in \mathfrak{gl}(V)$ is any nilpotent element, then the action ...
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What is the Jacobian for Sim(3) lie group action on 3D points ? (4d homogenous points)

I am coding up Sim(3) constraint types for a factor graph, and need to calculate the jacobian of the Sim(3) group action on 3D points. I am following the guide on http://ethaneade.com/lie.pdf ...
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Decomposition of SU(n) anticommutator

In $SU(N)$, the special unitary group, the algebra generators $T_a$ are hermitian and traceless. The structure constants are fixed with $[T_a,T_b]=i f_{abc}T_c$. In the fundamental representation of ...
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Closure relations of the cells in the Bruhat decomposition of the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$. How do we prove the closure relations between the cells, which ...
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question about infinitesimal transformations

Lawrence Dresner says this: (p. 10, Applications of Lie's Theory of Ordinary and Partial Differential Equations) Assume you have two infinitesimal group transformations: $$x'=x+\varepsilon(\lambda - ...
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Is the symplectic group $Sp(2n,\mathbb{R})$ simple?

Is the symplectic group $Sp(2n,\mathbb{R})$ simple? Wikipedia states that the Lie algebra $sp(2n,\mathbb{R})$ is simple. http://en.wikipedia.org/wiki/Table_of_Lie_groups However it only lists ...
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37 views

Is the exponential map for $\text{Sp}(2n,{\mathbb R})$ surjective?

For $\mathfrak{g} := {\mathfrak s}{\mathfrak p}(2n,\mathbb{R})$ and $G = \text{Sp}(2n,{\mathbb R})$, is the exponential map \begin{equation} \text{exp} : \mathfrak{g} \to G \end{equation} surjective? ...
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Why does the maximal compact subgroup of a Lie group inject into the compact form?

I've seen multiple sources state the following (without proof or reference), but I don't see why it's true. Let $G$ be a Lie group, and $G_u$ be a compact connected Lie group such that the ...
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1answer
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Lie algebra representations and tensor product decompositions.

Find the weights for $V_{L_1 - 2L_3}$, where $L_1, L_2, L_3$ are the weights for the standard representation of $\mathfrak{sl}_3 \Bbb{C}$ on $V \cong \Bbb{C}^3$. In order to find these weights, ...
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how to compute the norm of a root from the Cartan matrix?

As far as I understand, the Cartan matrix is associated with a unique semi simple algebra. How can we compute the norm of a root $\alpha$ from it since its components are invariant under rescaling? ...
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Decomposition into irreducibles of representations of semisimple Lie groups.

Let $G$ be a connected semisimple Lie group and $\mathfrak{g}$ it's Lie algebra. Then $\mathfrak{g}$ is semisimple. Let $V$ be a finite dimensional representation of $G$. Viewing $V$ as a ...
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Tangent space of matrix group is a Lie subalgebra

In my lecture today, we were covering matrix groups and Lie algebras. My professor made the statement that given any matrix group $G$, the tangent space of the group at the identity $T_{e}G$ is a Lie ...
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Finding the tangent space of a subgroup

My professor set the following question and I have an answer, though would like someone with more experience to cast a critical eye over the details as I don't necessarily trust my result! Define the ...
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How is the periodic structure of SO(n) reflected to its lie algebra so(n)?

An element of $SO(n)$ represents an rotation so that it must have identity with $2\pi$-like additional rotation. On the other hand, the elements of lie algebra $so(n)$ construct an noncompact vector ...
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How can the generators of subalgebra $\mathfrak g^{\sigma}$ of $\sigma$-stable elements be expressed through generators of Lie algebra $\mathfrak g$?

Let $\mathfrak g$ be the semisimple Lie algebra of type $D_{4}$. Let $\sigma$ be the 3-rd order automorphism of $\mathfrak g$ induced by the triality of $D_{4}$: $$ ...
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1answer
20 views

How to find the multiplicity of weight in a Verma module?

In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots. How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the ...
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The proportionality constant between the Casimir and the identity.

By Schur's Lemma, in any irreducible representation of a Lie algebra, the Casimir operator $J$ is proportional to the identity $Id$. How can we see that $J=j(j+1)Id$ for some natural number $j$ and ...
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Does the five lemma hold true for Lie algebras?

According to wikipedia, the Five Lemma is true in Abelian categories. But the category of Lie algebras is not Abelian. Then is the Five Lemma still true for Lie Algebras?
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The Lie algebra of special linear group of degree 2 over the set of complex numbers

I tried to compute the Lie algebra of $SL(2,\Bbb C)$. I wrote the followings: $sl(2,\Bbb C)$={$X\in M(2,\Bbb C)$: exp $tX$ $\in SL(2,\Bbb C)$}={$X\in M(2,\Bbb C)$: $det$ exp $tX$ =$1$}={$X\in M(2,\Bbb ...
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Find a vector field $\mathbb{Y}$ satisfying $L_{\mathbb{X}}\mathbb{Y}=\mathbb{Z}$

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $\mathbb{X}=(1,y)$. Let $\mathbb{Z}$ be the vector field on $\mathbb{R}^2$ given by $\displaystyle \mathbb{Z}(x,y)= \bigg( ...
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25 views

Find the lie derivative of a particular integral

Let $\mathbb{X}=(1,y)$ be a vector field on $\mathbb{R}^2$. Let $\Phi_t$ be the flow of $\mathbb{X}$. The flow of $\mathbb{X}$ I have calculated to be $\Phi_t(x,y)=(x+t,ye^t)$ Given a function ...