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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Branching rules without previous knowledge of the projection matrix?

Given a representation $R$ of some group $G$ one can find in many books and papers (e.g. page 96ff here) the decomposition under certain subgroups: $$ R= R_1 + R_2 + \ldots$$ This is often called a ...
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Inner Automorphism of Lie algebras in Terms of Roots and Weights?

An automorphism is a homomorphism of a group $G$ onto itself. For Lie groups this induces a Lie algebra $g$ automorphism, i.e. a map of the Lie alegbra onto itself that preserves the Lie bracket. An ...
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Maximal subalgebras of simple Lie algebras.

Does anyone know how I can access Dynkin's papers on the classification of maximal subalgebras of simple finite dimensional complex Lie algebras? V.V. Morozov also worked on this topic, how can I ...
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69 views

Orbit of a Weight Vector?

Given some element $\phi$ of a representation $R$ of a group $G$, the orbit $G(\phi)$ of $\phi$ is defined as the set $g \phi \ \forall \ g \in G$. We can write every element of a given ...
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27 views

Automorphism group of a lie algebra as a lie subgroup of $GL(\frak g)$

Let $G$ be a lie group with lie algebra $\frak{g}$. Let $Aut(\frak g)$ be the automorphism group of $\frak{g}$. Its clear to me that $Aut(\frak{g})$ $\subset GL(\frak{g})$ since any automorphism of ...
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33 views

Duality and tensor product of the Lie algebra

I would like to know how to compute the tensor product of the matrices below and how to deal with duality of vector spaces. The vector space I concern is the Lie algebra $\mathscr{sl_2}$ with basis ...
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1answer
34 views

How to proof that the $\mathbb{Z}$-span of weights of a faithful $L$-modul contains the root lattic?

Let $L$ be a semisimple Lie Algebra with root system $\Phi$ and base $\Delta$ of $\Phi$. Let $V$ be a finit dimensional, faithful $L$-modul with weights $\Pi(V)$. I am trying to show that the ...
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Adjoint action notations $\operatorname{ad}(X)(Y)$, $\operatorname{ad}(X)$ and $\operatorname{ad}_x(Y)$ are equivalent??

As the title says I'm a bit confused with these notations of adjoint action of Lie algebra on itself. Are these notations ($\operatorname{ad}(X)(Y)$, $\operatorname{ad}(X)$ and ...
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46 views

How to understand the definition of Killing form?

Define the matrix commutator $\text{ad}_X$ as $$\text{ad}_XY=[X,Y]=XY-YX$$ where $X,Y\in\mathfrak{g}$ and $\mathfrak{g}$ is the Lie algebra associated to Lie group $G$. Then on Lie group $G$, the ...
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35 views

Killing form question.

Studying Lie algebras (Cartan-Weyl basis) I've stumbled upon the expression of the Killing form: $g_{\rho\alpha}=c^\sigma_{\rho\tau} c^\tau_{\alpha\sigma}$. The question is are these ...
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Solving $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$

Given an arbitrary $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$? Update: ...
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46 views

Stabilizer subgroup of adjoint action

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...
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2answers
60 views

Representation of $sl(2,R)$.

I am interested in the unique (up to isomorphism) $5$-dimensional representation of the Lie algebra $sl(2,R)$. I understand that one can choose the module $V_4 = ...
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1answer
42 views

research in special function with Lie algebra

First of all, I don't know if this is the right place to ask about this. If not, please direct me somewhere I can get more help. I have to research in the field of special functions with a lie ...
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List of simple roots in the H-basis for various Lie algebras?

There are four usual bases one can use to express the roots and weights of a given algebra. The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$. The ...
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29 views

What exactly is lower central series of Lie algebra?

I've read several definitions of $LCS$ and derived series of Lie algebra, but I'm not sure if i get it right: In case of $LCS$, the relationship is given as $g_{k+1}=[g,g_k]$, does $``g_k"$ stand for ...
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32 views

Explicit Representation of the SU(N) Simple Roots in with redundant coefficents?

Commonly the simple roots for $SU(n)$ groups are given as $n$ dimensional vectors, although root-space is $n-1$ dimensional. The $SU(n)$ Wikipedia article explains: Here, we use n redundant ...
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23 views

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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1answer
25 views

Universal enveloping algebra as bialgebra

If $\mathfrak{g}$ is a Lie algebra (over $k$ ), then we can construct its universal enveloping algebra $U(\mathfrak{g})$. We can define $\Delta:U(\mathfrak{g})\rightarrow U(\mathfrak{g})\otimes ...
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19 views

$SO(n)$ algebra relations in the vector rep

The $\mathfrak{so}(n)$ algebra has some relations between generators always indicated as $$\left[T_{ij}, T_{kl}\right] = \delta_{ik}T_{jl} - \delta_{jk}T_{li} - \delta_{jl} T_{ik} + ...
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63 views

If $\mathfrak{g}=\bigoplus\mathfrak{g}_i$ is a semisimple Lie algebra, why does $\mathfrak{h}=\bigoplus\left(\mathfrak{h}\cap\mathfrak{g}_i\right)$?

There is this property about Cartan subalgebras that is not clear to me. Suppose $\mathfrak{g}$ is a semisimple Lie algebra. Then I know we can decompose it uniquely as ...
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How can one assume that an isomorphism of root spaces $\Phi\to\Phi'$ comes from an isometry?

By definition, if $\Phi$ and $\Phi'$ are root systems of the Euclidean spaces $E$ and $E'$, respectively, then an isomorphism $\Phi\to\Phi'$ is one that is induced by an isomorphism $E\to E'$ which ...
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gradient flow on $SU(n)$

Define the following cost functions $f_1, f_2 :SU(n) \rightarrow \mathbb{R}$ by $f_1(U) = Re \left( \text{Tr}\left(G^{\dagger} U \right) \right)$ and $f_2(U) = \left| \left( \text{Tr}\left(G^{\dagger} ...
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23 views

Root space $L_\alpha$ is completely contained in simple ideal?

I'm having trouble understanding a section in Humphrey's Lie algebras on page 74. Suppose $L$ is a semisimple Lie algebra which decomposes as a direct sum of simple ideals $L_1\oplus\cdots\oplus ...
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Weights system corresponding to reflected Dynkin diagram?

Given a set of weights corresponding to the $SO(10) Dynkin diagram How can I transform these weights into weights that correspond to the Dynkin diagram ?
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29 views

Calculating the Lie algebra representation of the regular representation on subspace of functions on $\mathbb R$.

Let $G = \mathbb R$ and let $\pi$ be the regular representation of $G$ on $L^2(\mathbb R)$, that is, $\pi(g)(f)(x) = f(x-g)$ for $g \in G$. Let $V = \{f \in \mathcal C_c^\infty | supp f \subseteq ...
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73 views

Invariant subspaces of Lie group vs invariant subspaces of Lie algebra

I am starting to study infinite-dimensional representations of Lie groups and I am wondering about the following: Let $G$ be a connected Lie group with Lie algebra $\mathfrak g$ and with a ...
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32 views

Map for roots of a Lie group to roots of a special subalgebra?

For regular subalgebras $h$ of some group's Lie algebra $g$, $$ h \subset h $$ the root system of the subalgebra is a subset of the root system of the original's group algebra. Subalgebras whose ...
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How to transform roots/weights from the simple root basis to the H-basis?

Often the roots and weights of some Lie algebra are written in terms of the simple root basis $$ r =(a_1,a_2,a_3,\ldots)=a_1 \alpha_1 + a_2 \alpha_2 + a_3 \alpha_3 +\ldots,$$ where $α_i$ denotes the ...
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49 views

What is the vector product $(x\wedge y)\wedge z$?

Here's an exercise from my book (exercise 10, chapter 2.1) Show that the three-dimensional vector space $V=R^3$ forms an associative algebra with respect to the operation $x\uparrow ...
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1answer
55 views

Anyone know a good standard reference for Lie group and Lie algebra facts?

I'm writing something and I need to refer to a mathematical fact; unfortunately I got it from Wikipedia, which does not source the specific piece of info! It relates to a choice of simple roots for ...
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145 views

Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
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35 views

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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46 views

Difference between infinitesimal motion and finite motion

I was reading an article about back ground of Killing's work by Thomas Hawkins from Historia mathematica 1980.In it Hawkin's says that,Killing was trying to generalise all types of space ...
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1answer
28 views

Cartan integers are preserved by isomorphism?

Suppose you have two roots systems $\Phi\subset E$ and $\Phi'\subset E'$ with bases of simple roots $(\alpha_1,\dots,\alpha_\ell)$ and $(\alpha_1',\dots,\alpha_\ell')$ such that the Cartan integers ...
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Identification of $H$ with $H^{*}$ relativ the Killing-form

Let $H$ be a maximal toral subalgebra of a semisimple Lie Algebra $L$. The identification of $H^{*}$ and H relativ the Killing-form says, that to $\phi\in H^{*}$ corresponds the unique element ...
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62 views

Show Lie bracket left invariant

I want to prove from the definition that the Lie bracket $[X,Y]$ of two left-invariant vector fields $X,Y: G \rightarrow TG$ where $G$ is a Lie group is again left-invariant. Left-invariance ...
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125 views

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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1answer
34 views

Lie algebra and Lie group cohomology, reference request

Who can give me a good reference (better if introductory/motivated) about Lie group cohomology, Lie algebra cohomology, and the link between the two? Thanks.
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65 views

equivalence of Lie group and Lie algebra intertwiner

I encountered this problem while working on my research. Let $G$ be a Lie group, and consider an intertwiner of the complex representations (possibly infinite-dimensional) $$ \pi:G\rightarrow ...
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1answer
30 views

How to find maximal dimension abelian subalgebra in finite Lie Algebra?

Is there any well known algorithm how to find maximal dimension abelian subalgebra in finite dimension Lie Algebra? If there is a built-in routine in some computer algebra system, it is the most ...
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1answer
31 views

Commutator ideal of reductive Lie algebra

I'm working through Fulton and Harris's book on Representation theory, and I've just done the exercise where I had to show: If $\mathfrak{g}$ is a reductive Lie algebra (defined as $Z(\mathfrak{g}) = ...
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36 views

Regarding proof of Theorem 3.3 in Humphreys (Lie alg. and Rep.)

Theorem 3.3. of Humphreys goes something like this: given a subalgebra $\mathfrak{g}$ of $\mathfrak{gl}(V)$ where $V$ is nonzero, finite-dimensional and $\mathfrak g$ consists of nilpotent ...
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1answer
50 views

References and suggestions about the elementary theory of Lie groups and Lie algebras

I am looking for suggestions on how to approach the field of Lie groups and Lie algebras. I am acquainted with both the elementary algebraic concepts, having studied from Bourbaki's "Algebra I-III", ...
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1answer
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The deconposition of $\mathfrak{so}(V \oplus V^*)$

Let $V$ be an n dimensional real vector space and $V^*$ be the dual vector space. We have a non degenerate inner product $(\centerdot,\centerdot)$ in $V\oplus V^*$ such that $(v+\xi , ...
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Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: ...
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$Z(\mathcal{L}^{(n)}) \subset Z(\mathcal{L})$ for solvable Lie algebras?

$X$ Banach space. $\mathcal{L} \in B(X) $ is solvable Lie Algebra. Then for some n, $\mathcal{L} \supset \mathcal{L}^{(1)}=[\mathcal{L},\mathcal{L}] \supset ...
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1answer
35 views

Every base of a root system arises as indecomposable positive roots of a regular element?

I'm confused about a line in the Theorem p48 in Humphrey's's book on Lie Algebras. He's proving that every base $\Delta$ of a root system $\Phi$ arises as the set of $\Delta(\gamma)$ of ...
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18 views

Adjoint map is Lie homomorphism

The Jacobi identity of a Lie algebra says that $ad: \mathfrak g \to End(\mathfrak g)$ is a derivation. I am a bit emberassed but what is the easieast way to see that for every $X \in \mathfrak g$, ...
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29 views

signature function of Weyl group element in LieArt

I am currently using LieArt Mathematica package for some calculations in Lie algebra, I am wondering if there is a way to know what is the signature of a Weyl group element, it seems the package can ...