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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2
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2answers
66 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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0answers
13 views

Lie algabra of R^n

Until now the only example of lie groups I have seen are subgroups of $GL_n$. Today I had the idea, that also $G=(\mathbb R^n,+)$ must be a lie group ($(\mathbb R^n,+)$ is a group with the ...
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1answer
30 views

In which course one learns Lie Group&Algebra and to which category of mathematics this subject belongs?

I'm a junior and i have never leanred this subject. I think "Lie Group&Algebra" is really deep and massive theory since the wikipedia page for it is quite long. Nevertheless, i'm not sure ...
0
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1answer
26 views

Exponents of a semisimple Lie algebra

I'd like to compute the exponents of a semisimple complex Lie algebra $\mathfrak{g}$. According to http://math.berkeley.edu/~theojf/LieQuantumGroups.pdf proposition 8.1.2.18, this amounts to ...
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0answers
11 views

How could we decompose anticommutator of representation matrices for a Lie algebra?

For commutator, we know that $[T^a,T^b]=if^{abc}T^c$, where $f^{abc}$ is the structure constant. But is there a similar formula for $\{T^a, T^b\}$? Thank you.
3
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2answers
37 views

Computing a Lie Bracket: General Questions

I'm asked to compute the following Lie Bracket: $\left [ -y \dfrac{\partial}{\partial x} + x\dfrac{\partial}{\partial y} , \dfrac{\partial}{\partial x} \right] $ on $\mathbb{R}^2$. Just writing it ...
7
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1answer
122 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
1
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1answer
38 views

What's wrong with my proof that reductive Lie algebras are semisimple?

If $L$ is a Lie algebra, $\text{Rad}(L)$ denotes its largest solvable ideal. Then $L$ is reductive if $\text{Rad}(L) = Z(L)$ (the center of $L$). An exercise in Humphreys asks: $L$ is reductive if ...
3
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0answers
48 views

What is the kernel of a Maurer-Cartan form?

The Maurer-Cartan form on the Lie group $Gl(n,\mathbb{R})$ is a one-form taking values in $\mathfrak{gl}(n,\mathbb{R})$ as defined in the link. It has a rather concrete "extrinsic definition" as ...
8
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1answer
271 views

Inscrutable proof in Humphrey's book on Lie algebras and representations

This is a question pertaining to Humphrey's Introduction to Lie Algebras and Representation Theory Is there an explanation of the lemma in §4.3-Cartan's Criterion? I understand the proof given there ...
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0answers
17 views

Chain rule to compute the Jacobian of a geometric transformation

This question is related to image alignment. I'm transforming some points in homogeneous coordinates then "de-homogenouzing" them. The transformation is a rigid-body transform in 3D parameterized by ...
0
votes
1answer
96 views

Lie group reps induced by Lie algebra reps

Let $G$ be a Lie group and $\mathfrak g$ its Lie algebra. Suppose that $\rho_\mathfrak{g}$ is a representation of $g$ on a vector space $V$. Is it true that the mapping $\rho$ from the identity ...
2
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0answers
33 views

Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
0
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0answers
16 views

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth.

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth. So where I am starting is by extending $Ad : G \rightarrow ...
0
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0answers
13 views

Double cover $Sp(1) \times SO(3) \rightarrow SO(4)$

I am working on understanding double covers at the moment. And I have come across a few double covers such as $Sp(1) \times Sp(1) \rightarrow SO(4)$ and $Sp(1) \rightarrow SO(3)$. And it seems to me ...
2
votes
2answers
55 views

Why does exponentiating the derivative yield the shift operator?

If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
2
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1answer
43 views

Representation theory and particle physics

Are there good books which explain clearly explain the connections between modern particle physics and representation theory of groups and lie algebras?
2
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1answer
21 views

Uniqueness of decomposition of $\mathfrak{sl}(2,\mathbb{C})$-modules

By Weyl's Theorem, I know that every $\mathfrak{sl}(2,\mathbb{C})$-module is completely reducible. I'm under the impression that, up to isomorphism, this decomposition is unique, and I would go about ...
4
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1answer
23 views

If the Lie algebra is a direct sum, then the Lie group is a direct product?

I am reading the corollary 21.6 in the book "Morse Theory" by John Milnor, but I've encountered a statement for which I have no ideas. Let $G$ be a simply connected Lie group with a bi-invariant ...
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0answers
16 views

Cartan subalgebra of product

i have a simple question what is the Cartan subalgebra of Lie algebra associated to the Lie group ?
5
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2answers
58 views

$\mathfrak{so}(n)$ has trivial center when $n\geq 3$

Is there a nice way to show that $$\mathfrak{so}(n)=\{A \in M(n,\mathbb{R}): A+A^t=0\} $$ has zero center for $n \geq 3$?
1
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1answer
33 views

Advice on proving a tricky inequality

Im a little out of my depth here and am not well versed in combinatorics. Im not sure if this problem is too hard to solve or if there exists well known results to prove it. Here is part 1 which might ...
0
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1answer
16 views

Kac-Moody root datum introductory text?

I have been given a project to describe the construction of the Lie algebra associated to a Kac-Moody root datum $D=(I,A,\Lambda, (c_i)_{i\in I}, (h_i)_{i\in I})$. I only know basic definitions: that ...
0
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1answer
17 views

Is the restricted enveloping algebra local?

Let $k$ be a field of characteristic $p>0$, and let $(\mathfrak{g},[\cdot,\cdot],(\cdot)^{[p]})$ be a finite-dimensional restricted Lie algebra over $k$. Let $u(\mathfrak{g})$ be the restricted ...
8
votes
5answers
196 views

Getting started with Lie Groups

I am looking for some material (e.g. references, books, notes) to get started with Lie Groups and Lie Algebra. My motivation is that I (eventually) want to understand the theory underpinning papers ...
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1answer
29 views

The trace as an integral over a sphere [duplicate]

Let $V$ be a real vector space of dimension $n$ and let $\langle \, \cdot\, , \,\cdot\, \rangle$ be an inner product on $V$. We can define a linear functional on the space of endomorphisms of $V$ by ...
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0answers
17 views

Is there a good list of accidental Lie algebra isomorphisms?

The Wikipedia page Exceptional isomorphisms contains some lie algebra isomorphisms. Is there a list more complete than that, especially including real algebras in low dimensions?
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0answers
31 views

A1 Lie algebra statement from Jürgen Fuchs' book “Affine Lie algebras and quantum groups”

On page 11, there is a statement saying that applying twice $ad_{E_\pm}$ to an arbitrary $$x = \xi_+E_+ + \xi_-E_- + \zeta H,$$ renders (obviously) $-2\xi_\pm E_\pm,$ the conclusion being that any ...
4
votes
2answers
114 views

How to prove that $B^\vee$ is a base for coroots?

Let $\Phi$ be a root system in a real inner product space $E$. Define $\alpha^\vee = \frac{2\alpha}{(\alpha, \alpha)}$. Then the set $\Phi^\vee = \{\alpha^\vee: \alpha \in \Phi \}$ is also a root ...
1
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1answer
37 views

Showing that an $\mathfrak{sl}(2,\mathbb{C})$-module is determined by eigenvalues of $h$

This question is essentially exercise 8.4 from the book "Introduction to Lie Algebras" by Erdmann and Wildon: "Suppose that $V$ is a finite-dimensional $\mathfrak{sl}(2,\mathbb{C})$-module. Show that ...
3
votes
1answer
52 views

How to deal with multiple representations of quaternions

I'm using a quaternion to represent the orientation in a kalman filter. My algorithm works fine until I rotate "upside down". I think this is because there seems to be multiple ways to represent the ...
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0answers
16 views

Any one any help our any specific book

The poisson bracket on Gr(Ug) {graded vector space over universal enveloping algebra} expressed in pol(g*) {dual of g is g*} is given by {f,g}(x)=x([df_x,dg_x]) where f,g are in pol(g*) and x in g*
1
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2answers
43 views

Invariance under conjugation, equivalent in Lie Group and Lie Algebra?

Is the following true? $ e^X Y e^{-X} = Y \Leftrightarrow [X,Y]=0$ . From right to left you can show it with a corollary from the Baker–Campbell–Hausdorff formula. But in the other direction? I ...
3
votes
2answers
93 views

“How many” matrices generate 2d Lie algebra (i.e. $[c_1,c_2]=k_1 c_1 +k_2 c_2$)?

Consider a pair of matrices $(c_1, c_2)$. The words "it generates the 2-dimensional Lie algebra", means that there exists a pair of scalars $k_1$, $k_2$, such that $$[c_1, c_2] = k_1 c_1 + k_2 c_2,$$ ...
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vote
0answers
24 views

Why is the restricted nullcone a variety?

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $(\mathfrak{g},[\cdot,\cdot],(\cdot)^{[p]})$ be a finite-dimensional restricted Lie algebra. Define the restricted ...
5
votes
2answers
88 views

traceless matrices

The fact that $\mathfrak{sl}_2(\mathbb{C})$ is a simple Lie algebra implies that every $2 \times 2$-matrix $A \in \mathbb{C}^{2\times 2}$ with $\mathrm{tr}(A) = 0$ can be expressed as a commutator of ...
0
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0answers
17 views

Infinite series of nested commutators

I'm trying to show the following: If $S_i$ are a set of three matrices such that $$ [S_i, S_j] = \epsilon_{ijk} S_k $$ then $$\exp\big( \alpha_i [S_i, \cdot]\big) S_j = (\exp (M) \vec{S})_j$$ ...
2
votes
0answers
21 views

Space of operators on function

Consider the following space of operators on function of $n$-variables $A= Span \{x_ix_j\ , x_i \frac{\partial}{\partial x_j} , \frac{\partial^2}{\partial x_i \partial x_j} , i,j=1,2,\cdots,n\}$. ...
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vote
2answers
38 views

The discription of abelian Lie groups

There is a problem in my problem sheet to discribe all abelian connected Lie groups (moreover this is the first problem and it should be rahter easy). First it is difficult to understand how this ...
2
votes
0answers
45 views

What is the Lie algebra of $G=\mathbb{R}$

The question is updated as following. 1. Let $(\Phi,L^2(R))$ be left regular representation of $\mathbb R$ given by $$ \Phi(g)f(x)=f(x-g). $$ It is unitary representation of $\mathbb R$. 2. For ...
0
votes
0answers
19 views

anticommutativity of lie algebras

With respect to the definition of Lie algebras, we note that the bilinearity and alternating properties imply anticommutativity i.e [x,y]=-[y,x] for all elements in Lie algebra. Now let L be a simple ...
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0answers
16 views

Combining infinitesimal generators of diferent dimensions

I am reading a paper about ways in which you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of $SU(3)\times{}SU(2)\times{}U(1)$. At a certain point, it starts considering ways of getting ...
3
votes
1answer
52 views

Inequality of Frobenius norm for skew matrices

Let $A$ be a complex skew-symmetric $n \times n$ matrix, that is, $A^T = -A$. Denote by $\|\cdot\|_F$ the Frobenius norm, that is, $\|B\|_F^2 = \text{trace}(B^*B)$. I would like to prove that $$ ...
0
votes
1answer
19 views

A question on the nilradical and the nilpotent ideal of $\mathfrak{p}\subseteq \mathfrak{gl}_n(\mathbb{C})$

Let $\mathfrak{p}\subseteq \mathfrak{gl}_n(\mathbb{C})$ be a parabolic algebra of a parabolic group $P\subseteq GL_n(\mathbb{C})$. What is the difference among the nilradical of $\mathfrak{p}$, the ...
10
votes
3answers
193 views

Translations in two dimensions - Group theory

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
6
votes
1answer
207 views

Dynkin diagram automorphisms and weights

Let $\sigma$ be a nontrivial Dynkin diagram automorphism of a finite-dimensional complex simple Lie algebra $\frak g$ (of type A, D or E) and let $\frak h$ be a Cartan subalgebra of $\frak g$. Let $I$ ...
2
votes
1answer
206 views

What physical meaning do the dimension of Wigner d-matrices have?

Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also ...
0
votes
1answer
73 views

What are the properties of this Poisson algebra?

I have the following (real) quantities (which are from a Classical Mechanics problem): $$A_1=\frac 1 4(x^2 +p_x^2-y^2-p_y^2 ) \quad A_2=\frac 1 2(x y +p_x p_y)$$ $$A_3=\frac 1 2(x p_y - y p_x )$$ ...
1
vote
1answer
33 views

For an element $x$ in an algebraic group $G$, why do we have $\mathscr{L}(C_G(x))\subset\mathfrak{c}_{\mathfrak{g}}(x)$?

I'm reading Humphreys' Linear Algebraic Groups, trying to understand the following argument found on the top of pg. 76. Let $G$ be an algebraic group over some field $k$, with $x\in G$. Let ...
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0answers
11 views

A canonical map Aut$_{\mathsf{Lie}_R}(\mathfrak{n} \rtimes_\pi \mathfrak{g}) \to$ Aut$_{\mathsf{Lie}_R}(\mathfrak{n})$

Let $\mathfrak{n}$, $\mathfrak{g} \in \mathsf{Lie}_R$ be two Lie algebras over a commutative ring $R$, s.t. $\mathfrak{g}$ acts on $\mathfrak{n}$ as a derivation: $\pi:\mathfrak{g} \to ...