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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Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
13
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361 views

Abelian Cartan subalgebras

If a Lie algebra is semisimple or reductive, its Cartan subalgebras are Abelian, and their elements semisimple. Are there non-reductive algebras with Abelian Cartan subalgebras all of whose ...
12
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183 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 ...
11
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161 views

Why are parabolic subgroups called “parabolic” subgroups?

I used to think that things called "parabolic" must have something to do with parabolas or their defining quadratic equations. In fact, terms like parabolic coordinate, parabolic partial differential ...
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124 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
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142 views

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 ...
7
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92 views

Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\...
7
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75 views

Solving a matrix differential equation

I am trying to solve: $\frac{d U_t}{dt} = Tr(G^{\dagger}U_t)G - Tr(U_t^{\dagger}G)U_t G^{\dagger} U_t$ Where $U_t \in SU(4)$ and $G \in SU(4)$ is given and constant. Is it possible to solve this ...
7
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177 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
7
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194 views

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 ...
7
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173 views

Non-degenerate bilinear forms of Lie algebra with a degenerate Killing form

Definition: A Lie algebra is defined by: $$ [e_a,e_b]={f_{ab}}^ce_c $$ The Killing form is $$ g_{ab}=-{f_{ac}}^d {f_{bd}}^c $$ Set-Up: The type of Lie algebra of our interests (found out during a ...
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141 views

Motivation?: Lie algebra and algebraic group Cohomology

This is just an a-priori question to get a motivational heuristic idea: If an algebraic group G (more generally, G an affine group scheme), is connected over an algebraically closed base-field k. ...
7
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188 views

Generating function for characters of representations

One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ ...
6
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88 views

Does solvability of Lie algebra have useful application in study of PDEs?

If certain Lie algebra is solvable then what difference this algebra would create in application point view for PDEs ? For example, in case of ODE of fourth order admitting three dimensional solvable ...
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84 views

A subspace is invariant by the Lie group if it is invariant by the Lie algebra

Let $G$ be a connected Lie group and $$\varphi:G\to \mathrm{GL}(V)$$ a representation on a finite dimensional real vector space $V$. Let $$\psi:\mathfrak{g}\to\mathrm{End}(V)$$ be the associated Lie ...
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93 views

How to calculate the Lie algebra of a neural network?

Define $F$ as the standard multi-layer feed-forward perceptron: \begin{equation} F(\mathbf{x}) = \Theta( W_1 \circ \Theta( W_2 \circ .... W_L(\mathbf{x}))) \end{equation} where $\Theta$ is the sigmoid ...
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813 views

Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
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219 views

An exercise in Serre's Lie algebra book

Let $k$ be a commutative ring. Prove that a Lie $k$-algebra $\mathfrak{g} = 0$ iff $U\mathfrak{g} = k$. Use the adjoint representaion. Here is my attempt at it: The only non-trivial statement is ...
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289 views

Root Systems of Lie Groups.

Let $G$ be a compact Lie group assumed to be a subgroup of $U(n)$. Also, let $T$ be a maximal torus of G. Then there exists a basis $\{v_1, \ldots ,v_d\}$ of the Lie algebra of $G$, $\mathfrak{g}$, ...
5
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48 views

Find the extended form of the group generated by an operator?

I tried to find the extended form of the group generated by the following operators. (I): The first operator $$A=z\frac{\partial }{\partial z}+1$$ To find the extended form of the group ...
5
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71 views

Pauli Matrices as Representations of Reflection Operators?

How does one derive, using, say, the Householder transformation $$ R(r) = (I - nn^*)(r),$$ the reflection representation of a vector $$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) + yR(\...
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90 views

Geodesics on Homogeneous Spaces

Consider a homogeneous space $G/\text{Stab}_p \cong M$ where $G$ is a compact Lie group active transitively on $M$ (a compact manifold). If $F$ is a Finsler Metric on $G$ which pushes forward ...
5
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72 views

Is every complex Lie algebra a complexification?

I'm wondering if every finite-dimensional complex Lie algebra can be written as a complexification of a real Lie algebra. At the vector space level, clearly every $\mathbb{C}^n$ is a complexification ...
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353 views

Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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59 views

Borel subalgebras inside the grassmannian

This is probably something standard and I just don't know where to look (so a reference would be just as appreciated as an answer), but... Let $\mathfrak{g}$ be a finite dimensional semisimple Lie ...
5
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96 views

duality for (co)homology of Lie algebras

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. What is the relationship between $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, $H^...
5
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48 views

Existence of product $\ast$ in a Lie algebra so that $[X,Y]=X*Y-Y*X$

I've been studying particle physics, and studying Lie algebra using physics text book doesn't give me enough information, so I'm asking my question here. Given a Lie algeba $\mathcal{A}$ where ...
5
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48 views

Are the two definitions of the BGG category equivalent?

Let $\mathfrak{g}$ be a finite dimensional complex semisimple lie algebra, the BGG category $\mathcal{O}$ is defined as the set of $\mathfrak{g}-$ module $M$ such that $M$ is finitely generated; $M$...
5
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281 views

Invariant bilinear forms on Lie algebras

Consider a (compact) Lie group $H$ that acts on its Lie algebra $\mathfrak h$ in the usual way, $x\mapsto gxg^{-1}$ for any $x\in\mathfrak h$ and $g\in H$. Suppose we are given a real symmetric ...
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132 views

— Cartan matrix for an exotic type of Lie algebra --

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra? For example, consider this Lie algebra: $$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = - f_{ik}{}^j Y^k \qquad\qquad [...
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85 views

Lie algebra associated to a linear form

Let $F$ be a finite dimensional vector space over a field $k$, if $f : F \to k$ is any linear form, I can define on $F$ a Lie algebra bracket by the following rule $$ [x,y]=f(x)y-f(y)x, $$ or in terms ...
5
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0answers
148 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let G be an affine group scheme, and Dist(G) its hyperalgebra. I am wondering what is the relationship between Dist(G) and G interms of Cohomology? Is there a cohomology theory for Dist(G), if so ...
5
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43 views

Does every element of $G_2 = \mathrm{Aut}(\mathbb{O})$ stabilize a quaternion subalgebra?

Has anyone heard of this result before? Let $\mathbb{O}$ denote the octonions and let $G_2$ denote its automorphism group (i.e. the 14 dimensional subgroup of $SO(7)$). Then any element of $G_2$ ...
5
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171 views

Irreducible representations of $\mathfrak{sl}_3\mathbb{C}$

I am working through the exercises in Fulton and Harris's Representation Theory, and am stuck on two on page 189. Let $\text{Sym}^2V$ denote the second symmetric power of the standard 3-dimensional ...
5
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109 views

Quantum Adjoint Action of the Coordinate Algebra on the Enveloping Algebra

As is well known, any Lie group $G$ has a canonical action on its Lie algebra $\frak{g}$, namely the adjoint action $Ad$. Firstly, let me ask, does this extend to an action of $G$ on its enveloping ...
5
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80 views

Are there any infinite dimensional subalgebras of the Witt algebra?

The Lie bracket of elements of the Witt algebra is given by: $[L_m,L_n]=(m-n)L_{m+n}$ Are there any infinite dimensional subalgebras of the Witt algebra that are not isomorphic to the Witt algebra ...
5
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125 views

Pullback of a 3-form to SU2

I have a left invariant 3-form, $\sigma$ on an simply connected Lie group, $G$ whose value at the identity is $\sigma=\langle[x,y],z\rangle$, where $\langle\cdot,\cdot\rangle$ denotes an invariant ...
4
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41 views

Why do Clifford bivectors represent the orthogonal Lie algebra?

It takes a long, painful, but straightforward calculation to see that the commutators of grade one elements $[\mathbf e_i,\mathbf e_j]$ of a Clifford algebra $\mathrm{Cl}(p,q)$ have exactly the same ...
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67 views

Open problems in Lie theory

I been studying lie theory for some time. Beside classification related problems what are some examples of open problems in the lie world? Especifically in the topological/differentiable structure of ...
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47 views

Multiplicities in weight diagram of representations of $\mathfrak{sl}(3,\mathbb{C})$

In the weight diagram of an irreducible (finite dimensional, complex) representation of $\mathfrak{sl}(3,\mathbb{C})$, there are 'rings' of weights in the shapes of triangles or hexagons. Is there an ...
4
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0answers
35 views

Radical of an infinite dimensional Lie Algebra

I am studying Lie Algebras and I just encountered the notion of radical $R$ of a finite dimensional Lie algebra $L$ over a field $F$, the maximal solvable ideal. Since the dimension of $L$ is finite, ...
4
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47 views

No local optima in quantum control?

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation: $\frac{d x(t)}{dt} = F(x,u)$ one can consider the ...
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25 views

Representation of a Kac-Moody algebra

Let $n$ be an integer $\geq 3$ and let $\mathfrak{g}$ be the Kac-Moody algebra with cartan matrix $C$ given by $C_{ij} = 2 \delta_{i,j} - \delta_{i,j+1} - \delta_{i,j-1} - \delta_{|i-j|,n-2}$. For ...
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111 views

Baker Campbell Hausdorff formula for unbounded operators

Baker Campbell Hausdorff formula says that for elements $X,Y$ of a Lie algebra we have $$e^Xe^Y=\exp(X+Y+\frac12[X,Y]+...),$$ which for $[X,Y]$ being central reduces to $$e^Xe^Y=\exp(X+Y+\frac12[X,Y])....
4
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54 views

Does a choice of measure on $\mathfrak{g}$ induce a measure on $G$?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. One can put a (left) Haar measure $\mu$ on $G$ and a Lebesgue measure $\lambda$ on $\mathfrak{g}$ which are both unique up to constants. My ...
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63 views

Basic application of Weyl-Character-Formula

(I did not find a solution of my problem in any forum so far. Sorry if it exists...) I am new to Lie-Algebras and representations and actually do not need the mathematical background... I need only ...
4
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57 views

Ideals of Lie-algebras

I am wondering whether the following claim is true: Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra and $V$ some vector subspace of $\mathfrak{g}$. Claim: $V$ is an ideal of $\mathfrak{g}$ ...
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172 views

How to understand Weyl chambers?

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane ...
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Recipe to compute dimension and decompose product of $SO(N)$ group representations

As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts: There is a correspondence between irreps ...
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197 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) \...