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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General form of a Lie Algebra

Given $G=\left\{A\in M_2(\mathbb{R})\mid A^\top XA = X\right\}$. Need to find the basis. Error in question
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1answer
180 views

$[y,x]=x$ in a non-abelian solvable Lie algebra

Proof that there exists non-zero elements $x, y$ in a solvable Lie algebra $g$ such that $[y,x]=x$. I have seen an answer from a lecture notes on google, but I can't find it now. Anyway, can someone ...
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119 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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2answers
139 views

What about Lie algebras over commutative rings?

It is just like the associative algebra over commutative ring (advanced linear algebra). It is a natural extension and can make the structure of Lie algebra more algebraic, but I find little book ...
2
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1answer
100 views

A solvable Lie-algebra of derived length 2 and nilpotency class $n$

Given a natural $n>2$, I want to show that there exists a lie algebra $g$ which is solvable of derived length 2, but nilpotent of degree $n$. I have seen a parallel idea in groups, but i can't see ...
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101 views

Question concerning semisimple Lie algebras

I'm currently solving a problem in Fulton's Representation Theory A first course and I'm not sure why a particular result is true. One part of the problem (exercise 14.15 if anyone is interested) ...
5
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1answer
112 views

How to prove $H^2(\mathfrak{g}, J(\mathfrak{g}))\neq0$, where $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$?

$\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra over a field $k$ with $\mathrm{char}k=0$. $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$. That is, the kernel of ...
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1answer
139 views

Are Lie algebras of non-isomorphic central simple algebras non-isomorphic?

Comments to my answer to this MO question, which is isomorphic to this MSE question, point out that I was tacitly assuming that the associated Lie algebras to non-isomorphic quaternion algebras over ...
5
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1answer
195 views

Lie Group Multiplication in Coordinates

I'm having a bit of trouble with the last bit of Problem 3.2 in Kirillov Jr.'s Introduction to Lie Groups and Lie Algebras. (3.2) Let $f: \mathfrak{g} \rightarrow G$ be any smooth map such that ...
2
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1answer
67 views

Motivation for filtrations on a group and associated Lie algebras

In his Lie algebras and Lie groups, the first non-trivial example of a Lie algebra that Serre gives is a graded Lie algebra associated with a filtration on a group. To me this construction looks both ...
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111 views

The Weyl Group of $F_4$

The Weyl Group of $F_4$ is of order $1152=2^{7} \cdot 3^{2}$. By Burnside's theorem the group is solvable. Is there a way to see solvability from the root system? Is it possible to see the order of ...
7
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1answer
155 views

Simplicity of $\operatorname{Der} \left(\mathbb F_p [x_1, \dots, x_n]/ (x_1^p,\dots, x_n^p )\right)$

I need to prove that the Lie algebra defined as: $W_{n} = \operatorname{Der} \left(\mathbb F_p [x_1, \dots, x_n ] / (x_1^p, \dots, x_n^p )\right)$, when $(x_1^p, \dots, x_n^p )$ is the ideal generated ...
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2answers
445 views

The fundamental group of Lie group

If $G$ is a compact Lie group whose Lie algebra $g$ has a trivial center, please show that the fundamental group of $G$ is finite.
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2answers
120 views

Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $

I've been trying to find a Lie algebra isomorphism $$\mathfrak o(4,\mathbb R)\cong\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $$ but haven't managed so far. I have written down the ...
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1answer
226 views

The symmetric algebra as a g-module

I'm quite sure that this question is not difficult, but I think that my understanding of the definitions is just not deep enough yet: Given a lie algebra $g$, and a $g$-module $V$ (or equivalently a ...
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1answer
193 views

The universal enveloping algebra of the trivial lie algebra

Given a commutative ring $k$ and a $k$-lie algebra $g$, I need to prove that the universal enveloping algebra $\mathcal{U}\left(g\right)=k\iff g=\left\{ 0\right\}$. One direction is very easy: If ...
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1answer
48 views

$O(3)$ after identifying certain rotations

suppose i have $O(3)$ as a group and then proceed to identify rotations on the same axis. That is, assuming an element in the simple component is written as $$ e^{s_i I_i } $$ where $I_i$ are ...
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1answer
77 views

counter examples for alleged sub-algebras

This is a homework question, in which we've got a bunch of kinds of subsets of a given Lie-algebra, and needed to decide wether these are sub-algebras, ideals, or non of the above. I have managed to ...
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0answers
107 views

Complete reducibility of finite-dimensional representations of $\mathfrak{sl}_2(\mathbb{C})$

By Weyl's theorem every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ is completely reducible, because $\mathfrak{sl}_2(\mathbb{C})$ is a (semi) simple Lie algebra. It seems there ...
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58 views

The form of a subgroup of $GL(n,K)$ when the derived group is of certain form

The famous Lie-Kolchin theorem in the theory of algebraic groups states: Let $G$ be a connected solvable subgroup of $GL(V)$, $0 \neq V$ finite dimensional. Then $G$ has a common eigenvector in ...
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1answer
68 views

Infinite dimensional $sl(2,\mathbb{C})$-modules

Let $\alpha$ be an arbitrary scalar in $\mathbb{C}$ and let $V(\alpha)$ be an infinite dimensional $\mathbb{C}$-vector space (with a countable basis). The formulas $h.v_i=(\alpha -2i).v_i$, ...
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1answer
166 views

Semisimple complex Lie algebra

Let L be a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L). (Der is short for derivation). In order to show that ad(L)=Der(L), the book says that it only need to show that the ...
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0answers
21 views

When does $C_G(s) \times Cl_G(s)s^{-1}$ equal $G$

I have read on James E. Humphreys' Linear Algberaic Groups If $G$ is an algebraic subgroup contained in $GL(n,K)$, and $s$ is a semisimple element of $G$, then $\mathfrak{g}$ has the ...
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2answers
624 views

Example of an associative cross product, any significance?

While trying to find cases that showed the cross product is not associative, I found some that were. I'm trying to show that $(\mathbf{A}\times \mathbf{B}) \times \mathbf{C} \ne \mathbf{A}\times ...
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1answer
75 views

If $U$ is a closed subgroup of $GL(V)$ consisting of unipotent elements, show that $\log x$ belongs to $\mathfrak{u}$

If $U$ is a closed subgroup of $GL(V)$ consisting of unipotent elements, show that $\log x$ $(x \in U)$ belongs to $\mathfrak{u}$. Here, for an unipotent element $x \in GL(V)$, $\log x = ...
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1answer
984 views

Casimir operator of a Lie Group .. how can i calculate ??'

given a Lie group with generators $ X_{i} $ how can i calculate the Casimir Generator ?? $ H= X_{i}X^{i} $ if possible with two examples please the Generator of traslation in 2 dimension $ P_{i} $ ...
4
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0answers
59 views

Proving $\mathscr L(C_G(H)) \subseteq \mathfrak c_{\mathfrak g}(\mathfrak h) = \{ \mathrm x \in \mathfrak g \mid [\mathrm x, \mathfrak h] = 0\}$

Let $H$ be a closed subgroup of the algebraic group $G$, $C = C_G(H)$. Prove that $\mathfrak{c} = \mathscr{L}(C_G(H)) \subseteq \mathfrak{c}_{\mathfrak{g}}(\mathfrak{h}) = \{ \mathrm{x} \in ...
2
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1answer
263 views

Complete reducibility of sl(3,F) as an sl(2,F)-module

I was reading the Weyl's theorem on the complete reducibility of a finite dimensional representation of semi-simple Lie algerba and wanted to apply the theorem to the following problem which was ...
0
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1answer
141 views

Existence of a basis of common eigenvectors

Suppose that L is a complex semisimple Lie algebra containing an abelian subalgebra H consisting of semisimple elements. I am wondering how to see that L has a basis of common eigenvectors for the ...
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1answer
334 views

Jordan Decomposition

Suppose that $x$ is a linear map from a finite dimensional vector space to itself, then $x$ has a unique Jordan Decomposition ($x=d+n$, $d$ is diagonalisable, $n$ is nilpotent). It's easy to show that ...
2
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1answer
81 views

Question about the parity of the ghost number operator in BRST quantization

Given a Lie algebra $[K_i,K_j]=f_{ij}^k K_k$, and ghost fields satisfying the anticommutation relations $\{c^i,b_j\}=\delta_j^i$, the ghost number operator is then $U=c^ib_i$ (duplicate indices are ...
7
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1answer
205 views

Three-dimensional simple Lie algebras over the rationals

Let $\mathfrak g$ be a three-dimensional $\mathbf Q$-vectorspace endowed with the structure of a simple Lie algebra. How many non-isomorphic such $\mathfrak g$ are there? Over the complex numbers, ...
0
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1answer
109 views

Express $ad_x$ in terms of the basis elements

I'm working with the set of trace zero matrices, $\mathfrak{sl}(V)\subseteq\mathfrak{gl}(V)$ of endomorphisms of a vector space $V$. The problem asks us to represent $ad_x, ad_y, ad_h$ in terms of ...
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0answers
81 views

Does triality survive in product Lie groups?

Look at the following diagrams of Lie groups ...
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3answers
239 views

Lie algebra of $GL_n(\mathbb{C})$

I would like to do Tao's exercise 6 (i) but before I can even attempt it I need to be clear about his terminology. Exercise 6 Show that the Lie algebra $gl_n(\mathbb{C})$ of the general linear group ...
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2answers
402 views

“Change of basis” from standard vector space to matrix Lie algebra, and its inverse

For matrix Lie groups, the exponential map is usually defined as a mapping from the set of $n \times n$ matrices onto itself. However, sometimes it is useful to have a minimal parametrization of our ...
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1answer
291 views

Definition of tangent space

Terry Tao defines tangent space here as equivalence classes of continuously differentiable curves $\gamma : I \rightarrow G$ where $I$ is an open interval. On the other hand, Wikipedia defines it as ...
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101 views

Why are root systems presented in this confusing way?

I quote Bjorner and Brenti, "Combinatorics of Coxeter Groups." We begin with a simple geometric lemma. Let $m \geq 3$ be an integer, let $\gamma = \pi/m$, and let $k, k'$ be real numbers ...
3
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1answer
455 views

Derivative of function including matrix logarithm

Is the following equation a first order approximation or incorrect for general matrix Lie groups? And what are the higher order terms? $$\frac{\partial}{\partial\mathbf x} (\log(\mathtt ...
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1answer
149 views

Is there any connection between fundamental group and diagram automorphism (for algebraic groups and root systems)

There is a conclusion on my textbook: All semisimple algebraic groups of type $F_4$ are isomorphic. I was confused because of the fundamental groups. Is it true that two semisimple algebraic ...
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0answers
100 views

Finite order automorphisms of Lie algebras

Let $\Gamma$ be a Dynkin diagram automorphism of diagram type $A_{2n}$ and let $\sigma$ be a non-trivial finite order automorphism of $\Gamma$. Let $g$ the Lie algebra associated to $\Gamma$ and ...
6
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1answer
343 views

Question about Weyl character formula

In the book of Humphreys, page 139, Weyl character formula is $$\left(\sum_{w\in W} \operatorname{sn}(w)\epsilon_{w\delta}\right) * \operatorname{ch}_{\lambda} = \sum_{w\in W} \operatorname{sn}(w) ...
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1answer
760 views

Properties of the longest element in a Weyl group

Let $w_0$ be the longest element in the Weyl group of a semisimple Lie algebra $\mathfrak{g}$. How does $w_0$ act on the simple roots $\{ \alpha_1, \ldots, \alpha_n \}$? If $L_{\lambda}$ is an ...
3
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1answer
155 views

Relation between a Lie group and Lie algebra representation for $W \otimes V$

We can define a representation of a Lie group and get the induced representation of the Lie algebra. Let $G$ act on $V$ and $W$, $\mathfrak{g}$ be the Lie algebra associated to $G$ and $X \in ...
3
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1answer
166 views

Cartan matrices in different books

Cartan matrices in the books of (1) Humphreys (page 59) and of (2) Carter (page 82 -- 83) are different. Moreover, they are not transpose of each other. Which one is correct? Thank you very much. ...
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1answer
295 views

tensor product of modules of Lie algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra and $M, N$ be two modules of $\mathfrak{g}$. Is it true that $M \otimes N \cong N \otimes M$? If $\mathfrak{g}$ is replaced by other algebras, $M \otimes ...
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2answers
378 views

irreducible highest weight modules

Let $\mathfrak{g}$ be a simple Lie algebra. Let $M_{\lambda}$ be the Verma module over $\mathfrak{g}$ of highest weight $\lambda$ and $L_{\lambda}$ be the irreducible $\mathfrak{g}$-module of highest ...
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0answers
325 views

Elementary proof of the third Lie theorem

I would like to understand a (not well known) proof due to G.M; Tuynman, of the third Lie theorem, which asserts that for any given finite dimensional Lie algebra $\mathcal{G}$ there exists a (simply ...
2
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1answer
129 views

Group algebras.

Given a group $G$. Let $\mathbb{C}$ be the complex field. Then $\mathbb{C}G$ is the set of linear combinations of elements of $G$. Addition and multiplication are defined as usual. We can also think ...
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2answers
264 views

Reference request: group theory

Currently I'm studying differential geometry and PDEs - so I often meet the use of groups. I also studied symmetries methods for solutions of differential equations but the connection between Lie ...