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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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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7 views

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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19 views

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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42 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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30 views

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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18 views

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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48 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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26 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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95 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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27 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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57 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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33 views

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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40 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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48 views

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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26 views

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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22 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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40 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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21 views

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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44 views

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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46 views

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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49 views

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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19 views

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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26 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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22 views

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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42 views

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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28 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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38 views

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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30 views

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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25 views

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) = ...
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26 views

Complexifying Lie group actions

In Atiyah-Pressley's paper Convexity and Loop Groups, it is claimed that the Bruhat cells $C_\lambda$ are invariant under the natural $S^1 \times T$-action and that the authors will prove this claim. ...
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42 views

Irreducible Representations of Nilpotent Lie algebras

By Lie's theorem all irreducible representations of a solvable Lie algebra over $\mathbb C$ are one dimensional. What are all irreducible representations of a nilpotent Lie algebra ?
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What series/publishers would be interested in publishing a novel mathematics book?

What series/publishers would be interested in publishing a novel mathematics book for a select readership? In other words, I'm looking for a publisher smaller than Springer, who thought my book ...
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74 views

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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18 views

Normal subgroup invariant under $\text{Ad}_g$

Denote by $G$ a Lie group with corresponding Lie algebra $\text{Lie}(G)$. There the three maps inner automorphism/conjugation: $\text{Int}_g = L_{g^{-1}} \circ R_g \in \text{Aut}(G)$, $\text{Ad}_g ...
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$H^q(\mathfrak{g},K;V)$ is equal to $Ext_{\left(\mathfrak{g},K\right)}^q\left(\mathbb{C},V\right)$?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $K$ be a closed subgroup of $G$ with corresponding Lie subalgebra $\mathfrak{k}$. Let $V$ be a $\left(\mathfrak{g},K\right)$-module. Then, I ...
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22 views

Clarification of Definition: Free Algebra

I need some clarification on the definition of free algebra. Here is an extract from Lie Algebras and Lie Groups by Jean-Pierre Serre: I am somewhat confused about the definition of free algebra. ...
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Reference request: classification of simple Lie groups and simple real Lie algebras

I am trying to understand the classification of simple Lie groups and the theory of highest weights for semisimple Lie groups by first understanding the case for complex Lie algebras, then relating to ...
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60 views

Lie Bracket of vector fields on Lie group

Let $H$ be a Lie group and $\mathfrak{h}$ its Lie algebra. Given a smooth function $v: H \to \mathfrak{h}$, define the vector field $\bar{v} : H \to TH$, $h \mapsto d(R_{h})_{e} v(h)$, where $R_{h} : ...
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52 views

Computing Jacobian of error function using Lie Algebra

First off all, I hope this is the right place to ask, as it is a computer vision problem, but I'm specifically asking about the mathematical part of it. I am currently implementing the ICP (Iterative ...
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20 views

coboundary operators in relative lie algebra cohomology

I am starting to read relative lie algebra cohomology. We define the coboundary operator $d$ from $Hom_K(\wedge^q\mathcal{g}/\mathcal{k}, V)$ to $Hom_K(\wedge^{q+1}\mathcal{g}/\mathcal{k}, V)$ as ...
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How to tackle a research journal - level course in Lie Theory and Representation Theory?

I am taking a course in Lie Theory and Theory of Representations this year, where starting from the second lecture, Lie Theory is heavily bundled with Theory of Representations. It is pretty much a ...
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91 views

John Lee's book question about symplectic group

Exercise 12-14 in John Lee's book, An introduction to Smooth Manifolds, reads as follows: The real symplectic group is the subgroup $Sp(n, \mathbb{R}) \subset GL(2n, \mathbb{R})$ consisting of ...
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47 views

Lie algebra associated to an arbitrary discrete group

I read somewhere that there is a classical (due to Philip Hall?) construction of a Lie algebra associated to any discrete group $\pi$ which is obtained from filtration quotients of the descending ...