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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How to show that the structure constant of SU(3) is invariant?

So suppose $f_{ijk}$ is the antisymmetric structure constant of SU(3), and $D^8_{ij}(g)$ is the matrices of 8-dimensional adjoint representation of SU(3), then how to show that ...
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116 views

Do the classical Lie algebras all satisfy $XM + MX^T = 0$?

I'm working on a homework assignment in which part of the question statement says that each of the classical Lie algebras can be described as the set of all matrices $X \in gl(n,\mathbb{C})$ ...
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70 views

orbit of a Dynkin diagram automorphism

Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$. What is a set of representatives of the orbits of $\Delta$ under $f$ ? Thanks,
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313 views

Length of root strings

Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra $g$ of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the length of an $a$-string through $b$ is at most ...
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12 views

What is a $\mathbb{Z}$-form of an algebra?

A homework problem I have is to describe the Lie algebra associated to a Kac-Moody root datum $\mathcal{K} =(I,A,\Lambda ,(c_i)_{i\in I},(h_i)_{i\in I})$ as well as to describe the universal ...
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19 views

Root spaces of Lie Algebras — semisimple vs. general

(I am mainly following the notation of Roger Carter's Lie Algebras of Finite and Affine Type). Letting $L$ denote a (finite-dimensional) Lie algebra with roots $\Phi$ and Cartan subalgebra $H$, we ...
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16 views

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth.

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth. So where I am starting is by extending $Ad : G \rightarrow ...
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14 views

Double cover $Sp(1) \times SO(3) \rightarrow SO(4)$

I am working on understanding double covers at the moment. And I have come across a few double covers such as $Sp(1) \times Sp(1) \rightarrow SO(4)$ and $Sp(1) \rightarrow SO(3)$. And it seems to me ...
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16 views

Cartan subalgebra of product

i have a simple question what is the Cartan subalgebra of Lie algebra associated to the Lie group ?
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31 views

A1 Lie algebra statement from Jürgen Fuchs' book “Affine Lie algebras and quantum groups”

On page 11, there is a statement saying that applying twice $ad_{E_\pm}$ to an arbitrary $$x = \xi_+E_+ + \xi_-E_- + \zeta H,$$ renders (obviously) $-2\xi_\pm E_\pm,$ the conclusion being that any ...
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16 views

Any one any help our any specific book

The poisson bracket on Gr(Ug) {graded vector space over universal enveloping algebra} expressed in pol(g*) {dual of g is g*} is given by {f,g}(x)=x([df_x,dg_x]) where f,g are in pol(g*) and x in g*
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17 views

Infinite series of nested commutators

I'm trying to show the following: If $S_i$ are a set of three matrices such that $$ [S_i, S_j] = \epsilon_{ijk} S_k $$ then $$\exp\big( \alpha_i [S_i, \cdot]\big) S_j = (\exp (M) \vec{S})_j$$ ...
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20 views

anticommutativity of lie algebras

With respect to the definition of Lie algebras, we note that the bilinearity and alternating properties imply anticommutativity i.e [x,y]=-[y,x] for all elements in Lie algebra. Now let L be a simple ...
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16 views

Combining infinitesimal generators of diferent dimensions

I am reading a paper about ways in which you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of $SU(3)\times{}SU(2)\times{}U(1)$. At a certain point, it starts considering ways of getting ...
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11 views

A canonical map Aut$_{\mathsf{Lie}_R}(\mathfrak{n} \rtimes_\pi \mathfrak{g}) \to$ Aut$_{\mathsf{Lie}_R}(\mathfrak{n})$

Let $\mathfrak{n}$, $\mathfrak{g} \in \mathsf{Lie}_R$ be two Lie algebras over a commutative ring $R$, s.t. $\mathfrak{g}$ acts on $\mathfrak{n}$ as a derivation: $\pi:\mathfrak{g} \to ...
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34 views

Ideals in the unitary group

What would be examples of one-dimensional ideals in the lie algebra of the unitary group? Moreover, how would one show that it is in the tangent space of the center of the unitary group and that the ...
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22 views

Adjoint representation as a member of group of automorphism

For $L$ as a finite dimensional Lie algebra , the adjoint mad $\mathrm{ad}:L \rightarrow \mathrm{End}(L)$ when $L$ is finite dimensional then $\mathrm{End}(L)$ is isomorphic to general linear group ...
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22 views

Basis of Witt algebra

The Witt algebra $W(n,m)$ is defined as the set of element $\{\sum f_j D_j$ such that $ f_j ∈ A(n,m)\}$ with usual Lie bracket. I am a bit confused about basis for $W(n,m)$? What is the meaning of ...
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20 views

Bilinear form on the space of smooth complex valued functions.

Let $G$ be a Lie group and $h$ be the Hermitian bilinear form on smooth complex valued functions then how can we define bilinear form on the space of smooth complex valued functions.
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33 views

A question on Lie Algebra

This is an exercise from 'Lie Algebras in Particle Physics' by Howard Georgi, Ex.6.B. Suppose that the raising lowering operators of some Lie Algebra satisfy: $[Eα,Eβ]=NE(α+β)$, where the $α,β$ are ...
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59 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
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22 views

Help with proof in Fulton and Harris

On page 488, let $H$ be any element of $\mathfrak{g}$ such that the generalized null space $\mathfrak{g}_0(H)$ has minimal dimension. Then consider $X\in\mathfrak{g}_0(H)$, and the decomposition $X = ...
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20 views

Why $\widehat{G^{\mathbb{C}}}$ can be identified with the space of highest weights

Let $G$ be a compact connected Lie group and $G^{\mathbb{C}}$ be the complexification of Lie group $G$ and we denote $\widehat{G^{\mathbb{C}}}$ the set of isomorphism classes of irreducible rational ...
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27 views

Weyl Character Formula to find $M_\lambda(\mu)$

In Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the dimensions of the ...
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33 views

Equivalent representations of $\mathfrak{sl}_2$

Hello I have a question about the equivalence of two representations of the Lie algebra $\mathfrak{sl}_2$. The first representation is $(ad,\mathfrak{sl}_2)$ the adjoint representation with map ...
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14 views

Invariants of exterior power of Lie algebras

Let $\mathfrak{g}$ a simple finite dimensional Lie algebra, and consider $$\bigwedge(\mathfrak{g}\oplus\mathfrak{g}).$$ Let $\{e_i\}$ and $\{f_i\}$ be dual basis of $\mathfrak{g}$ with respect to the ...
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19 views

Invariants of representation of simple Lie algebras.

Let $\mathfrak{g}$ a finite dimensional simple Lie algebras and let $V$ a representation of $\mathfrak{g}$ such that $$V=\bigoplus_{i,j\in I}(L(\mu_i)\otimes L(\mu_j)).$$ Where $L(\mu_i)$ is the ...
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35 views

holomorphic Lie group automorphisms of the complex Heisenberg group preserving the lattice

I don't understand Lie group,but I need an elementary result about it,so I ask for help for a simple question. Take a complex Heisenberg group $G$ $$ \left( \begin{array}{ccc} 1 & z_1 & ...
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33 views

Irrep dimensions of non semisimple Lie algebra

I'm mostly interested in Lie algebra "numerology". The book "Birdtracks" and the website http://www-math.univ-poitiers.fr/~maavl/LiE/form.html answered me everything on irrep dimensions for semisimple ...
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22 views

Constructing Lie algebra from the associative algebra.

Show that any associative algebra $A$ can be made into Lie algebra by taking $[x,y]=xy-yx$ for any $x,y \in A$. The way I would tackle it. $\circ$ Clearly $A$ is a vector space as it is an ...
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28 views

Heisenberg algebras and their ideals

We know that Heisenberg Lie algebra is a Lie algebra $H(2m+1)$ with basis $v_1, \ldots , v_{2m}, v$ and the only non--zero multiplication between basis elements is given by $[v_{2i-1}, v_{2i}] = - ...
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162 views

Short proof of Jacobi identity for the Poisson bracket — is this valid?

I've been trying to make a short proof for the Jacobi identity for the Poisson bracket on phase space. My idea goes like this, we know the following: $$ \frac{\mathrm{d}}{\mathrm{dt}} \{f,g\} = ...
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28 views

References request about exponentials in Lie algebras.

I saw two formulas about Lie algebras. Let $G$ be an algebraic group over $k$ and $\mathfrak{g}$ its Lie algebra. For any $x \in \mathfrak{g}$, $a \in k$ and $g \in G$, we have $$ g \exp(ax) g^{-1} = ...
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34 views

Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
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36 views

Extend to a homomorphism.

My question is regarding a step in "p-Automorphisms of Finite p-Groups" by Evgenii I. Khukhro (p. 117 line 7) and would like some response to my argumentation/understanding of it. p-Automorphisms of ...
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14 views

A property on $SU(2^N)$

Is there a simple formula for the following: $\frac{d}{d \epsilon} \exp(\lambda^k G_k + \epsilon G_j) |_{\epsilon = 0}$ where ${G_k}$ are the standard generators of $SU(2^N)$ given by N-fold tensor ...
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62 views

sl2(C) does not have a nontrivial 1-dimensional central extension?

How can I show lie algebra sl2(C) does not have a nontrivial 1-dimensional central extension?
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20 views

irreducible representations of lie algebras

We have the following criterion for the irreducibility of a Lie algebra representation (we work with $L$-modules here). Let $L$ be a Lie algebra, $V$ a finite dimensional vector space, and let $L ...
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25 views

parabolic subalgebra

Let $G$ be a semisimple lie group, a parabolic subgroup of $P$ is a connected subgroup that contains a conjugate of $B$, (which $B$ is Borel subgroup of $G$) then I can not see why lie algebra of $P$ ...
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11 views

Is the matrix of every set of base vectors of $\Bbb{C}^n$ symmetric?

The book "Theory of Lie Groups" by Chevalley says A linear endomorphism $\alpha$ of $C^n$ is determined when the elements $\alpha e_i=\sum\limits_{j=1}^n a_{ji}e_j$ are given. There corresponds ...
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40 views

definition of derived algebra $[L,L]$ of a Lie algebra $L$

Definition of derived algebra of a Lie algebra $L$ is given by linear span of commutators $[x,y]$ for $x,y \in L$. but here why do we take linear span and why cant we just consider collection of all ...
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25 views

Kunneth-like formula for Lie algebra cohomology

Let $\mathfrak g$ be a complex Lie algebra and $V$ a (not necessarily finite dimensional) representation. Suppose $\mathfrak g = \mathfrak g_1 \oplus \mathfrak g_2$ as a vector space (not Lie ...
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90 views

Exercise 1.17 from Introduction to Lie Algebra and Representation Theory by Erdmann and Wildon

Let $V$ be an $n$-dimensional complex vector space and let $L = \mathfrak gl(V )$. Suppose that $x ∈ L$ is diagonalisable, with eigenvalues $λ_1, . . . , λ_n$. Show that $\operatorname{ad} x ∈ ...
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46 views

isomorphism between $so(3) \oplus so(3)$ and $so(4)$

Could you show me explicit isomorphism between $so(3) \oplus so(3)$ and $so(4)$? (in the basis of skew-symmetric matrix)
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27 views

How to prove a property of taking Lie derivatives of a tensor field

If A is a tensor field of type $(^{k}_{l})$on N and $\phi:M \to N$ is a diffeomorphism, we define $\phi^{*}A$ as follows. If $v_{1}, \dots , v_{k} \in M_{p}$, and $\lambda_{1},\dots, \lambda_{l} \in ...
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93 views

Schur's lemma and Invariant subspaces of direct sums of irreducible representations

There is a corollary to Schur's lemma which says that : If $V$ is a finite dimensional irreducible complex representation of a group G or Lie algebra and $\phi :V \rightarrow V$ is an intertwining ...
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78 views

Kernel of the Lie bracket $[,]\colon\wedge^2\mathfrak g\to\mathfrak g$

I believe the following is probably well-known, but so far I couldn't find the answer by myself: Let $\mathfrak g$ be a real (finite-dimensional) Lie algebra, and $\wedge^2\mathfrak g$ its second ...
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41 views

A $\mathbb{Z}$-graded Lie superalgebra from a Lie algebra

Let $\mathfrak{h}$ be any $\mathbb{K}$-Lie algebra. We set $\mathfrak{g}_{-1}=\mathfrak{h}$ (as vector space), $\mathfrak{g}_0=\mathfrak{h}$ and $\mathfrak{g}_1=\mathbb{K}$ (or any one dimensional ...
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32 views

A question about Lie Groups

I'm just trying to complete the argument. The aim is to prove that $SO(3)$ does not admit a left-invariant flat Lorentz metric. Well, suppose it does. Then $R(x,y) = 0$ for all $x$ and $y$ in it's Lie ...
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48 views

Roots and Root vectors of $sp(4,\mathbb{R})$.

I found that the cartan subalgebra of $sp(4,\mathbb{R})$ is the algebra with diagonal matrices in $sp(4,\mathbb{R})$. Now to find out the roots I need to compute: $$[H,X]=\alpha(H) X$$ For every ...