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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14 views

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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67 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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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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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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48 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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28 views

$Z(\mathcal{L}^{(n)}) \subset Z(\mathcal{L})$ 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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20 views

How do I know the length of a Weyl group element from its Weyl orbit result

How do I know the length of a Weyl group element from its Weyl orbit result? For example, I know that $[2,2]$ under $s_1s_2s_1$ transforms into $[-2,-2]$,but given the result, how can I tell ...
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40 views

Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
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Computing the Cohomology of Lie groups

In Bredons "Topology and Geometry" [Chapter V, section 12] the following theorem is proven: If $G$ is a compact connected Lie group its $k$-th cohomology $H^k(G,\mathbb{R})$ is isomorphic to the ...
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Lie algebra operations from lie group

According to wikipedia, if $G$ is a closed subgroup of $GL(n, \mathbb{R})$ then the Lie algebra of $G$ can be thought of informally as the matrices $m$ of $M(n, \mathbb{R})$ such that $1 + εm$ is in ...
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The algebra of $W$-invariant polynomial funktions sl2

Let $L=\mathfrak{sl}(2,\mathbb{F})$, $H$ a borel subalgebra, $\Delta=\{\alpha\}$ a base of the corresponding root system and $W$ the Weyl group. Let $\lambda=\frac{1}{2}\alpha$ be the fundamental ...
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34 views

Universal enveloping algebra of sl2

I am currently trying to proof, that $x-1$ is not invertible in the universal enveloping algebra $\mathfrak{U}(\mathfrak{sl}(2,\mathbb{F}))$ of $\mathfrak{sl}(2, \mathbb{F})$, but I still struggle ...
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20 views

Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$? In the case of $G = SL_2$, we have $\mathbb{C}[SL_2] = \langle a,b,c,d\rangle / (ad-bc-1 )$ and ...
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How can Clebsch-Gordan Decompositions be combined?

In section 4 of this paper the authors use a given list of Clebsch-Gordan coefficents for the $27 \otimes 27$ of $E_6$ from an old paper and combine it with their own list of Clebsch-Gordan ...
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A Lie Algebra $L$ is reductive iff it is completely reducibile as an $\operatorname{ad}_L(L)$-module

Given a Lie Algebra $L$ we say it is reductive if $\operatorname{Rad}L=Z(L)$. How can we prove that $L$ is reductive iff it is an $\operatorname{ad}_L(L)$-module completely reducibile? Suppose $L$ ...
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22 views

Question about the theorem of highest weights

I have some confusion from reading Theorem 7.3 in Sepanski's Compact Lie groups and would appreciate it if someone could clarify. In part (e) the book says "for $w\in W$, $wV_\lambda=V_{w\lambda}$, ...
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43 views

Unknown proof Lie Algebra

I have a calculation where I do not know what it actually shows. I think it tells me that for right invariant vector fields, the commutator is again right invariant. Maybe somebody here could help me ...
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Dimension of Conjugacy class in $SU(n)$

Consider $D \in SU(n)$ ($n$ a multiple of 4), a diagonal matrix with values $\pm 1$ on the diagonal and with trace 0 (only possible for $n$ a multiple of 4). There are $n \choose n/2$ such matrices. ...
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Why do we need the Dynkin Basis to compute Branching Rules?

Given a representation $R$ of some Lie algbra $g$, we can compute the corresponding representation $R'$ (in general reducible) for some subgroup Lie algebra $ g \supset g'$ by utilizing the weights in ...
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53 views

De Rham cohomology of $T^n$ using Künneth formula and Chevalley-Eilenberg theorem.

I want to calculate $H^*(T^n)$ with ring structure using both of these methods. Künneth formula gives $$ H^p(T^n)=H^p(S^1\times T^{n-1})=\bigoplus_{i+j=p}H^i(S^1)\otimes H^j(T^{n-1}) $$ for each ...
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29 views

Why does a simple coroot $\alpha^{(i)}$ correspond to a Cartan subalgebra element $H^i$?

I read here that a simple coroot $\alpha^{(i)}$ corresponds to a Cartan subalgebra element $H^i$ and don't understand why this should be the case. Roots are the weights of the adjoint ...
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101 views

Right invariance of Casimir (Laplacian)

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. The Casimir element $\Delta\in \mathfrak{zu}(\mathfrak{g})$, considered as an operator on $C^{\infty}(G)$ is right invariant, that is, $\Delta ...
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30 views

Complex irreducible representation of solvable lie algebra

How can one infer from the Lie's theorem (in terms of existence of a common eigenvector) that a complex irreducible representation of a solvable lie algebra has dimension 1? What I know is that one ...
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22 views

Subspaces of Lie algebras

The Lie correspondence is well understood. For 'nice enough' Lie groups $G$ (with Lie algebra $\mathfrak{g}$) every sub-group $H < G$ has a Lie algebra $\mathfrak{h} < \mathfrak{g}$ given by ...
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34 views

Is there a general method to calculate the generators of the subgroups of $\textrm{GL}(n,F)$?

I know this might be a very bad/broad question, but after going through a few practice problems for finding linearly independent generators for some of the easier subgroups of $\textrm{GL}(n,F)$ ...
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61 views

Showing that the Witt algebra $W(1)$ is isomorphic to $\mathfrak{sl}(2,\mathbb{k})$

As the title suggests, I need to show that the Witt algebra $W(1)$ with basis $\{e_i \, | \, -1 \leq i \leq p-2\}$ where $e_k=t^{k+1}\frac{\mathrm{d}}{\mathrm{d}t}$ with Lie bracket defined by ...
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How to write $\mathfrak{su}(3)$ Lie algebra as a sum of two subspaces?

Let $K,F\subset\mathfrak{su}(3)$ be subspaces, such that $K \oplus F =\mathfrak{su}(3)$, and $K$ has a $\mathfrak{su}(2)$ structure. How can we show that $[K,K] = K$ (i.e., commutator of any two ...
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42 views

Why generator in Lie Algebra is defined as the coefficient in taylor expansion of map

Booth defines the infinitesimal generator of a lie group (denote the manifold it defines by $M$) using flow $\theta_t(p)$ by calculatng the limit (mainly the derivation for $f$ in each point $p\in M$) ...
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23 views

The negative of a vector field and its flow

I have a relatively short question about vector fields. Let $G$ be a Lie Group, and $X$ a smooth vector field on it. If its flow is $\left\{\phi_t\right\}$ what is the corresponding flow for $-X$? ...
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Special linear lie algebra relative with Angular Momentum

Let have the special linear algebra $\operatorname{Sl}(2,\mathbb F)$ ,which is the set of $ 2 \times 2$ matrix with trace zero. I have to prove that the lie algebra $ g=\operatorname{Span}\{ ...
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Where does the $-k\delta$ part of the expression for weights come from?

Consider the affine Lie algebra with the Cartan matrix $$ \left(\begin{array}{cc} 2 & -2\\ -2 & 2 \end{array}\right) $$ Let $\omega_{0}$ be the zeroth fundamental weight, $\alpha_1$ the first ...
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23 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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41 views

Dominant Weight

I am reading a paper which begin by a reminder about root system associated to a simple lie algebra $\mathfrak g$. let $\mathfrak h\subset \mathfrak g$ a cartan subalgebra. Question 1: It says that ...
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21 views

Questions about the indivisible imaginary root in affine root system.

I am reading the paper. On page 5, $\delta$ is defined as the indivisible imaginary root in $\widehat{\Delta_+}$. $\Lambda_0 \in \widehat{\mathfrak{h}^*}$ is the unique element satisfying $\langle K, ...
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Why is 1/2+1/2 in the weight space for SO(5)

Let's consider $\mathfrak{so}(5)$ as the Lie algebra of $\mathrm{SO}(5)$, where the symmetric bilinear form is $x_1y_5+\cdots +y_1x_5$. Then the maximal torus is given by $$\left(\begin{array}{cccccc} ...
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Induced Lie Algebra Representation, Left invariant vector fields and more…

The following is an excerpt from a proof in John Lee's Introduction to Smooth Manifolds I am struggling to understand. I would appreciate if someone was able to help me with whatever it is I am ...
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Maximal tori in Lie vs algebraic groups

If $G$ is a Lie group, we define a maximal [Lie] torus in $G$ to be a maximal connected compact abelian Lie subgroup of $G$. These guys correspond to Cartan subalgebras of $\mathfrak{g}=Lie(G)$. If ...
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22 views

An injection of Weyl groups

I've shown, quite accidentally, that Weyl group of $F_4$ injects into the Weyl group of $E_6$ as the subgroup of elements normalizing a maximal torus $T^4$ of $F_4$. One might a priori expect other ...
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cohomology ring of Lie algebras: multiplication

If $\mathfrak{g}$ is a Lie $R$-algebra, then the Chevalley-Eilenberg complex defines the cohomology modules $H^k(\mathfrak{g})$. If $H^\ast(\mathfrak{g})=\bigoplus_kH^k(\mathfrak{g})$, then the ...
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centre of universal enveloping algebra for specific algebras

Let $\mathfrak{h}_{2n}$ be the Heisenberg Lie algebra, i.e. the Lie algebra with a basis of $\{p_1,\ldots,p_n,q_1,\ldots,q_n,c\}$ where $$[Pi, Pj ] = [Qi, Qj ] = [Pi, C] = [Qi, C] = [C, C] = 0, [Pi, ...
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27 views

Importance of Jordan-Chevalley decomposition

What are the uses of Jordan-Chevalley decomposition in the classification of semisimple Lie algebras? I used it to prove that the restriction of the killing form to a maximal toral subalgebra is non ...
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Determining whether a Lie algebra is also a complex Lie algebra

I am trying to learn Lie theory. In the following I will share my thoughts. Please, can you check my work for correctness and point out any mistakes to me? I am trying to determine whether ...
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Equivalences between categories $\mathcal{K}^b(\text{Injectives})$ and $\mathcal{D}^b(\mathcal A)$ if $\mathcal{A}$ has enough injectives

I have the following question: Let $\mathcal{A}$ be a abelian category and $\mathcal{I}$ be the full subcategory of injective objexts of $\mathcal{A}$. Assume that $\mathcal{A}$ has enough ...
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30 views

Good reference for The Differntiable Slice Theorem

I am looking for a book that will give me a good proof of The Differentiable Slice Theorem - Suppose a compact Lie group $G$ acts smoothly on a manifold $M$. Then every orbit has a $G$-invarient ...
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Hochschild-Serre spectral sequence for not normal subalgebra

I am trying to understand lemma 2.26 from http://www.math.ru.nl/~solleveld/scrip.pdf I am coserned about calculation of $E^{p, q}_1$. If $\mathfrak{h}$ is Lie ideal than everything is fine. But here ...
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Why does $det(R) = +1$ imply right handed frame?

Let $R$ be a rotational matrix in $SO(3)$ so it satisfies $R^TR = I$ Solvng for $det(R^TR) = (det(R))^2 = 1$ yields two solutions Why does $det(R) = +1$ mean that the frame is a right handed frame? ...
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33 views

Commutators of Schur polynomials of Lie algebra elements

Question: Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...
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Lie point symmetry of KdV.

I'm asked to consider the 1-param. group of transformations generated by $V = \dfrac{\partial}{\partial u} + \alpha t \dfrac{\partial}{\partial x}$, which easily enough yields $g^{\epsilon}(x,t,u) = ...