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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Straight forward derivation of the bch formula?

Im doing a project on rigid body dynamics and need to derive the bch formula, anyone know a simple yet complete derivation?
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Exponentials of Representations of Lie Algebras

Assume G is a lie group and g is its lie algebra. Consider a representation of G : D:G->End(V). Then there is a corresponding representation of g : d:g->End(V). My question is, when you can express ...
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General differentials operators (Grothendieck definition) and polynomial rings

Let $A$ be an algebra over some field $\mathbb{k}$. A linear map $f:A\to A$ is said to be a differential operator of an order $\le n$ if for all $a_0,a_1,\ldots a_n\in A$ we have ...
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on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, maybe semisimple); call its Cartan subalgebra $\mathbf t \subset \mathbf g$ and Weyl group $W$. Why does the construction $\mathcal ...
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1answer
27 views

What is the natural action of $\mathfrak{sl}(4,\Bbb{C})$ on $\wedge^2 \Bbb{C}^4$?

What is the natural action of $\mathfrak{sl}(4,\Bbb{C})$ on $\wedge^2 \Bbb{C}^4$? We know that $\wedge^2 \Bbb{C}^4$ is generated by $\{e_1 \wedge e_2, e_1 \wedge e_3, e_1 \wedge e_4, e_2 \wedge e_3, ...
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Motivating Complexification of Lie Algebras? [migrated]

What is the motivation for complexifying a Lie algebra? In quantum mechanical angular momentum the commutation relations $$[J_x,J_y]=iJ_z, [J_y,J_z] = iJ_x, [J_z,J_x] = iJ_y$$ become, on ...
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1answer
80 views

Differential in Lie groups

I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Where $\phi(t)$ is a one-parameter subgroup of the Lie group ...
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Partial derivatives on Manifolds

Let $F : A \times B \to C$ be a map of smooth manifolds. Define the following maps ("partial derivatives"): $E_1 F: TA \times B \to TC$ $E_1 F(a,v,b) = D_a F(-,b) v $ where $v \in T_a A$ $E_2 F: A ...
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An alternative proof for the units of $U_q(\mathfrak{sl}_2)$ using Ore extensions.

I would like to establish what the set of units are in the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$. First, I recall the definition of the quantized enveloping algebra- throughout the ...
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1answer
113 views

Matrix exponential converse. Baker-Campbell-Hausdorff

I am currently reading about the Baker-Campbell-Hausdorff formula and in a textbook on Lie Algebras, he shows that if $$[X,[X,Y]] = 0 \quad \text{ and } [Y,[X,Y]] = 0$$ then $$e^{Xt}e^{Yt} = e^{Xt ...
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1answer
41 views

What is a simple lie algebra?

What is a simple lie algebra? What should I be thinking of when I come across these? What is a good example or two that I should keep in the back of my mind at all times? I know they are useful, but ...
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49 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 $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, ...
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1answer
57 views

Lie algebras and the Killing form.

The Killing form is defined by $K(x,y) = \text{tr}(\text{ad} x, \text{ad} y)$, right? In this lecture, we assume that $\{x_1, ... , x_n\}$ is a basis for $g$ and $\{y_1, ... ,y_n\}$ is a dual basis ...
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1answer
40 views

Correspondence between one-parameter subgroups of G and TeG

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
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1answer
29 views

Quaternions as a Lie algebra, its derivations

Let $\mathbb{H}$ be the algebra of quaternions. It can be proven that each derivation $D:\mathbb{H}\to \mathbb{H}$ is inner that is of the form $\mathrm{ad}x$ for some $x\in \mathbb{H}$. I am to prove ...
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gradation of lie algebras

Let $ H(n,m) $ for $n=2r$ and $K(n,m)$ for $n=2r+1$ be hamilton and contact lie algebras over finite fields. $ H(n,m) $ is a graded subalgebra of $W(n,m)$ with length $s=\sum _{i=1} ^{2r} (p^{m_i} ...
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1answer
30 views

Two definitions of left-invariant vector fields of a Lie group

I am reading these lines from a text which shows why the bracket of two left-invariant vector fields is also a left-invariant vector field. But cannot easily get one of the lines. Let $L_a$ be the ...
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1answer
22 views

Matrices of subrepresentations and quotient representations.

Suppose that $V$ is a $5$ dimensional representation (with generators $\{y_1, ... , y_5\}$ of the lie group $\mathcal{g}$, with the lie algebra homomorphism $\rho: \mathcal{g} \rightarrow ...
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24 views

Study of Lie Algebra

I saw this statement in an old note which I was trying to study: "A bilinear multiplication is not necessarily associative". I'm at pains however to see this as I attempt the proof!
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nilpotent radical of lie algebra

I need an example of computing nilpotent radical of lie algebra over finite field. In other words, I want to know how can I find the basis for nilpotent radical of Lie algebra over a field of positive ...
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Computations of common isometry groups, $O(n)/O(n-1), SO(n)/SO(n-1), U(n)/U(n-1)$, etc?

On wikipedia, some of the common isometry groups are given: $S^{n-1}\cong O(n)/O(n-1)$, $S^{n-1}\cong SO(n)/SO(n-1)$, etc. Is there a reference where some/any of these groups are computed? I'm just ...
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141 views

What does the logarithm of the identity look like on a Lie group?

Let $G$ be a compact, connected Lie group with identity element $e$ and $\mathfrak g$ its Lie algebra. Consider the set $$ L=\{A\in\mathfrak g\setminus\{0\};\exp(A)=e\}. $$ The most descriptive name ...
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1answer
45 views

Different definitions of Casimir element

I read about the Casimir element just recently and I found it a bit difficult to wrap my mind around the definition(s). In fact, I have seen two different definitions. For concreteness, let ...
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32 views

Why do Ad(K) orbits in the $-1$ eigenspace of a Cartan decomposition intersect the Weyl chamber once?

Let $G$ be a semisimple Lie group and let $\frak p\oplus t$ be a Cartan decomposition of $\frak g$ and $K$ the connected subgroup with Lie algebra $\frak t$. Choose a maximal abelian subalgebra ...
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42 views

Identifying a Lie algebra from its universal enveloping algebra

Its been a while since I've worked on Lie algebras and I can't remember how to approach this problem: How do I identify the lie algebra (up to isomorphism) associated to a certain universal ...
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1answer
31 views

Reflection in terms of simple reflections

Suppose $\beta=\sum_{i=1}^ka_i\alpha_i$, where $\alpha_i$ are simple roots. Is there any easier way to write the reflection corresponding to $\beta$ say $s_{\beta}$ in terms of $s_{\alpha_i}$'s. I ...
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2answers
75 views

How to show $\exp(tX)\exp(tY)=\exp(t(X+Y)+tR(t))$ with $\displaystyle \lim_{t\to 0} R(t)=0$?

Let $X\in GL(n, \mathbb R)$. The exponential of $X$ is the matrix given by $$\exp(X)=\sum_{n=0}^\infty \frac{X^n}{n!}.$$ I need some help for showing the following result: ...
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Derivative of exponential map

Somehow I've gotten myself confused trying to take the derivative of the exponential map on $\mathfrak{so(3)}$. For vector $\theta$, $\delta \theta$, and $p \in \mathbb{R}^3$, define $$R(\theta, p) ...
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1answer
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derivative of composition of rotations

Let $\theta$ and $\psi$ be two vectors in $\mathbb{R}^3$. I want to compute $$\nabla_{\psi} \log \left( e^{[\theta]_\times}e^{[\psi]_\times}\right)$$ Where $[v]_\times$ is the skew-symmetric ...
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1answer
42 views

Is there a natural Lie bracket for $\mathfrak X (M) \times C^\infty(M)$ (pairs of vector fields and smooth functions)?

Space of smooth vector fields $\mathfrak X(M)$ on a manifold $M$ has a structure of Lie algebra with the bracket being a commutator of two vector fields. Does cartesian product $\mathfrak X (M) ...
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37 views

To check d^2= 0 in the standard complex of Lie superalgebras.

For a Lie superalgebra $\mathfrak{g}$ and a $\mathfrak{g}$-module $V$ we can define the cohomology $H^i(\mathfrak{g}, V)$ with coeffiecient in $V$ to be the cohomology space of the following complex: ...
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1answer
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Matrix Lie algebras

I gave an answer to Is there a group between $SO(2,\mathbb{R})$ and $SL(2,\mathbb{R})$? which was not popular. Meanwhile, i found myself at a loss when wishing to explain why a matrix Lie group had, ...
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24 views

reflection(reflection) = rotation

Lel $\alpha$ and $\beta$ be two distinct simple roots in a root system $\Phi$. How to prove that i) $S_{\alpha} S_{\beta}$ is a rotation in $\mathbb{R}\Phi$ ii) Composition of two reflection is a ...
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1answer
41 views

simple contact algebra

Let suppose we have contact lie algebra K(3,(2,2,2)) over GF(3), according to the book "Modular lie algebras and their representations" from Helmut Strade, K(3,(2,2,2)) is simple Lie algebra ( it ...
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A refernce about Cartan matrix

There exist an approach to "Cartan Matrix" in Carter's book "Finite groups of Lie type, conjugacy classes an complex characters" p.23, which seems be different to other definitions of Cartan matrix I ...
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38 views

Differential operators on the polynomial ring

Let $A$ be a commutative algebra over complex numbers. If $a\in A$ we define $m_a$ to be a linear map which sends each $x$ to $ax$. The zero map $A\to A$ is said to be a differential operator of an ...
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1answer
32 views

Killing form on some Lie algebra $L$ is zero. Is $L$ necessarily nilpotent?

I've solved an exercise in Humphreys that said: Show that if Lie algebra $L$ is nilpotent, then its Killing form is zero. I'm wondering is the opposite true? In Humphreys, we work mainly with ...
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1answer
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How to show that $\alpha_{i_p}(s_{i_{p-1}} \cdots s_{i_1}(h)) = (s_{i_1} \cdots s_{i_{p-1}}(\alpha_{i_p}))(h)$?

Let $i_1, \ldots, i_p$ be integers and $\alpha_i$ be simple roots and $s_i$ be simple reflections in a Weyl group of type A. I checked some examples and it seems that $$\alpha_{i_p}(s_{i_{p-1}} \cdots ...
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24 views

Dimension of maximal tori

Let $G$ be a compact Lie group. $T$ $-$ its maximal torus. Is there a simple reasoning to show that dimensions of $T$ and $G$ have the same parity? I am sorry if this quesion is for children, but ...
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Confusion in infinite-dimensional Lie algebra notes

I've been going through these notes and everything was pretty much fine until I saw line labeled (5.1) on page 6. What kind of object is $z^\alpha (dz)^\beta$? How does exactly differential operator ...
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Is the exponential map of GL(n,C) holomorphic?

Let $GL(n, \mathbb{C})$ be the complex general linear (Lie) group consisting of all invertible complex $n\times n$ matrices, and $gl(n,\mathbb{C})\cong C^{n^2}$ be its Lie algebra. The exponential map ...
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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 ...
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1answer
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How to find the Lie algebra of a specific subgroup of a product Lie group

My question is about finding the Lie algebra of a specific Lie group. Start with a Lie group $G$, with normal Lie subgroup $C \unlhd G$. Then define the following subgroup $\hat{G} \leq G \times G$: ...
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1answer
39 views

Action of Symmetric Group on Lie Polynomials with GAP

Let $L$ be the free Lie Algebra, freely generated by $x_1,x_2, \ldots, x_n$. Let $f$ be a polynomial in $L$ and $\sigma \in S_n$, how to do $\sigma$ act on $f$ in GAP? That is $$\sigma f(x_1, \ldots, ...
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2answers
106 views

How to show a matrix can't be written as exponential?

How can I show the matrix $$A = \left( \begin{array}{c c} -2 & 0 \\ 0 & -1 \\ \end{array} \right)$$ can't be written as $A = exp(a)$? I've tried to write A like $$A = \left( ...
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3answers
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Indecomposable representations of Lie algebra

Let $\mathfrak{g}$ be the nonabelian $2$-dimensional complex Lie algebra. It can be generated by two independent vectors $e_1,e_2$ such that $[e_1,e_2]=e_1$. Thus, $\mathfrak{g}$ is solvable and it ...
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1answer
33 views

Finding a basis for $sp(4,\mathbb{C})$ and related basis.

Let $$L = so_4(\mathbb{C})= \{x \in End(\mathbb{C}^4)|^txS + Sx = 0 \} \text{ where }S = \left(\begin{array}{cc} 0 & I_2 \\ -I_2 & 0 \end{array}\right)$$ Letting $x = \left(\begin{array}{cc} ...
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Questions about the bracket

In the map $\phi : L \mapsto \mathfrak {U}(L) $, where $ L $ is a lie algebra and $\mathfrak {U} $ is a universal enveloping algebra of $ L $. (1) Is the following relation true? If $[xy]=z$ in $ L ...
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2answers
42 views

Direct sum decomposition of weight spaces and relation to Tensor products.

There are 3 parts to the question that I am trying to understand, and while it is not homework it seems instrumental in decomposition modules into weight spaces and their relation to tensor products. ...
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1answer
40 views

Some examples for Lie algebras

I need some small examples for Lie algebras over finite fields ( GF(2) or GF(3)) including some simple Lie algebras and some others which are not simple. And I would be thankful if anyone could give ...