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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Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with ...
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1answer
75 views

Specific help in showing that Poisson Bracket is part of this Lie Algebra

Given in this exercise is the following set: $U = \{f(z) = z^TCz\ \vert\ C \in \textrm{Mat}_{2n}(\mathbb R),\ C^T=C\}$ is a Lie Algebra with $\left\{\cdot , \cdot \right\}$ where $$ \left\{f,g ...
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1answer
234 views

What is the relationship between semisimple lie algebras and semisimple elements?

A Lie algebra $\mathfrak{g}$ is said to be semisimple if its radical is zero. An element $x \in \mathfrak{g}$ is said to be semisimple if $\text{ad} x$ is diagonalizable. A complex semisimple Lie ...
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1answer
79 views

Non split extension of lie algebra?

Is there a Lie Algebra $\mathfrak g$ so that the extension $0\xrightarrow{}\mathfrak h\xrightarrow{}\mathfrak g\xrightarrow{}\mathfrak q\xrightarrow{}0$ does not split, i.e. $\mathfrak g$ is not a ...
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59 views

Invariants of representation theory of Lie groups

How to compute the determinant of a representation of an element of the special linear group? How do I argue that it doesn't change? (@Marek: @rschwieb: Yes well, given one represenation (with ...
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2answers
105 views

Action of $\mathfrak{sl}({V})$ in tensor spaces

what is the natural action of $\mathfrak{sl}({V})$ in tensor spaces ?
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1answer
182 views

Special Case of Lie-algebra

Suppose $\Bbb{R}^3$ with $[u,v]=u\times v$, thus the cross product of $u$ and $v$ and suppose also $\mathfrak{so}(n)$, the space of skew symmetric $n\times n$-matrices with $[a,b]=ab-ba$. Then i have ...
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2answers
204 views

Orthonormal basis of Cartan subalgebra relative to Killing form

I'm trying to understand a step in a proof: Let $\mathfrak{g}$ be semi-simple (finite dimensional) Lie-algebra over $\mathbb{C}$, $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra and let ...
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79 views

$x$ regular $\Leftrightarrow$ $x$ is in exactly one CSA

Here's a statement and a proof given in a Lie Algebra course (in the tutorial): Let $L$ be a semisimple Lie algebra over a field $F$ with $\text{char} F=0$. Let $x\in L$ be a semisimple element. ...
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1answer
208 views

Dimension of Lie algebra according to root system

I was wondering how is it possible to find the dimension of a semi-simple lie algebra $L$ if its corresponding root system is (lets make it simple) of type $B_2$. We can find the number of roots and ...
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2answers
119 views

Nilpotent Lie Group that is not simply connect nor product of Lie Groups?

I have been trying to find for days an non-abelian nilpotent Lie Group that is not simply connect nor product of Lie Groups, but haven't been able to succeed. Is there an example of this, or hints to ...
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2answers
100 views

The Lie algebras $\mathfrak{o}_3(\mathbb{C})$ and $\mathfrak{sl}_2(\mathbb{C})$

The adjoint representation of $\mathfrak{sl}_2(\mathbb{C})$ under a natural basis, it is given by $$\text{ad}: \mathfrak{sl}_2(\mathbb{C})\to\mathfrak{gl}_3(\mathbb{C})$$ ...
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36 views

Lie Derivative in Projective Hilbert Space

In considering a projective Hilbert space, $P(H)$, for linear maps (tensors) of vectors in the space, $A^{a}_{b}v_{a}=u_b$, is there a natural definition for the Lie Derivative for such linear maps? ...
2
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2answers
315 views

Length of root strings is at most 4

Let $\Phi$ be a root system. In his Introduction to Lie algebras and Representation Theory, J. Humphreys proves that if $$\beta-p\alpha,\dots,\beta,\dots,\beta+q\alpha$$ is the $\alpha$-root string ...
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1answer
54 views

Example ideal of $\mathfrak{sl}(2,\mathbb{C})$

I need an example about ideal from lie algebra $\mathfrak{sl}(2,\mathbb{C})$ except trivial ideal and $\mathfrak{sl}(2,\mathbb{C})$ itself, can someone help me? I try to make ideal except trivial ...
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62 views

Finite-dimensional Lie algebra as a scheme

Kindly asking for any hints about the following questions: Suppose $k$ is an algebraically closed field of characteristic zero and $g$ is a finite-dimensional Lie algebra over $k$. Then $g$ is ...
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1answer
132 views

Invariant inner products on infite-dimensional representations

Let $G$ be a compact group and let $V$ be it's continuous representation. It is well known that if $V$ is finite-dimensional, then there is an $G$-invariant inner product on $V$. I haven't found a ...
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2answers
211 views

When does the $\mathfrak g$-invariance of the symplectic form imply $G$-invariance?

Let $G$ be a Lie group with Lie algebra $\mathfrak g$, and let $M$ be a smooth manifold. Suppose $G$ acts on $M$, $G \to \text{Diff}(M)$. This naturally induces an action $\mathfrak g \times M \to M$ ...
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1answer
62 views

Getting a derivation from a path of automorphisms of an algebra

These representation theory notes leave the following claim to the reader: Recall that a derivation on an algebra is a map $d$ such that $d(ab)=d(a)b+ad(b)$. If $A$ is a finite-dimensional algebra ...
4
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42 views

Is it possible to further simplify the product of three exponentials $e^A e^B e^C$ when $[A,C]=kB$ (k is a scalar)

The background is calculation of the little group elements of Poincare group for massless particles. I start with a bunch of exponentials of operators, and the end goal is to crunch them into the ...
3
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1answer
60 views

Nilpotence of Lie Algebra

I am trying to show that if $L$ is a Lie algebra and $L/Z(L)$ is nilpotent than $L$ is also nilpotent. Can someone please help me? I tried to first show by induction: $(L/Z(L))^k=L^k/Z(L)$. Is it ...
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3answers
151 views

what is the difference between $\mathrm{SL}(2)$ and $\mathfrak{sl}(2)$?

Regarding $\mathrm{SL}(2)$ and $\mathfrak{sl}(2)$, may I know what is the difference between them? I understand that $\mathrm{SL}(2)$ is the set of $2\times 2$ matrices with determinant $1$. ...
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117 views

$\Delta \subset \Phi$ is a base in a root system imples $\Delta^\vee \subset \Phi^\vee$ is a base in a root system [duplicate]

(the notation here is compatible with J.E. Humphrey's "Introduction to Lie Algebras and Representation Theory") Let $\Phi \subset E$ be a root system. Let $\Delta \subset \Phi$ be a base. I already ...
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1answer
360 views

Finding All Irreducible Representations of $SO(3)$

I've read that one may prove that all irreducible representations of $SO(3)$ are tensor product representations of the fundamental representation (or tensor product representations of the spin 1/2 ...
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1answer
49 views

How to write down a presentation of a Lie algebra if we know a set of generators?

How to write down a presentation of a Lie algebra if we know a set of generators in matrix form? For example, for $sl_2$, if we know $e=(0, 1; 0, 0)$, $f=(0, 0; 1, 0)$ , $h=(1, 0; 0, -1)$, how to ...
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2answers
274 views

Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?

At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint? (..a related query: ...
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1answer
82 views

Lie algebra 3 Dimensional with 2 Dimensional derived lie algebra #2

I read the book of Karin Erdmann and Mark Wildon: "An introduction to Lie algebras". In page 22 they say that: If $\dim (L) = 3$, $\dim (L') = 2$ then (a) $L'$ abelian and (b) $\operatorname{ad} x ...
2
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1answer
143 views

establishing an isomorphism

I request help with this is a question from Introduction to Lie algebra by Erdmann and Wildon. The question asks to show that show that $so(4,\mathbf{C})\cong sl(2,\mathbf{C}) \oplus ...
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2answers
513 views

Is every skew-adjoint matrix a commutator of two self-adjoint matrices

I'm looking to solve some matrix equations. One of the equations involves a commutator, so my question is as follows: let $A$ be a skew-self-adjoint, traceless matrix, does the equation $[X,Y] = A$ ...
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1answer
65 views

Indecomposable L-module

I have the following exercice which I have be trying to solve: Let L be a Lie algebra and $r:L\rightarrow gl_3(F)$ a representation of L such that $im(r)=t_3(F)$ (the upper triangular matrices). Show ...
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1answer
46 views

Lie product of a two subalgebras

Let V and W be subalgebras of a Lie algebra $\mathcal{L}$. I want to show that $[V,W]$ is not always a subalgebra of $\mathcal{L}$.
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1answer
265 views

The Universal enveloping algebra of a finite dimensional Lie algebra is Noetherian.

If $\mathfrak{g}$ is a finite-dimensional Lie algebra, then it is very known that the Universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ is a Noetherian ring. What is the simplest way to ...
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110 views

Submanifold of a Lie group - tangent space

Let $G$ be a compact Lie group and $H, H' \leq G$ Lie subgroups. Consider the set $M = H' \cdot H = \{h\cdot h' \ \vert \ h \in H, h' \in H'\}$. Is it possible to describe explicitly the tangent space ...
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2answers
217 views

Does the proof of the Poincare-Birkhoff-Witt theorem need the Jacobi identity?

The title says it. Suppose I have a vector space $V$ equipped with a bilinear bracket such that $[x,y]=-[y,x]$, and define the universal enveloping algebra $U$ as usual: namely the tensor algebra on ...
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0answers
75 views

Dimension of Abelian Lie Algebras

I've tried to answer this question, but I need some help. What is the possible dimension of irreducible representations of Abelian Lie Algebras? I think it is always one, but I am not sure. Thank ...
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1answer
95 views

A question about Lie algebras corresponding to Lie groups and algebraic groups

Lie groups and algebraic groups both correspond with Lie algebras, which are by definition the left invariant vector field. But the topology of Lie groups and algebraic groups are different. Are their ...
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1answer
101 views

Sanity check on spider web calculation

For fun I began reading an interesting online paper I found, Spiders for rank 2 Lie algebras, and on page $5$ we have the following calculation, akin to a tensor product expansion via bilinearity: ...
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181 views

When is the adjoint representation self-dual?

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...
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2answers
195 views

Lie algebra 3 Dimensional with 2 Dimensional derived lie algebra

i read in mark wildon book , an introduction to lie algebras, in page 22 say that : Suppose that dim $L$ = 3 and dim $L'$ = 2. We shall see that, over $\mathbb{C}$ at least, there are infinitely many ...
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1answer
56 views

preserves eigen spaces?

"Let $H_0=\begin{pmatrix}i&0\\0&-i\end{pmatrix}$, suppose $A\in SU(2)$ commutes with $H_0$, it must preserves each eigen spaces for $H_0$, eigen spaces for $H_0$ are just $\mathbb{C}e_1$ and ...
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1answer
112 views

Weyl group of $\mathfrak{sl}(2,\mathbb{C})$

$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{gl}(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the ...
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72 views

Is there any Lie algebra that is not constructed from an associative algebra

I see in Wikipeida that every Lie algebra is either constructed from an associative algebra by defining: $[x,y]=xy-yx$, or a subalgebra of a Lie algebra thus constructed. Where can I find a proof? ...
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1answer
58 views

Different orderings for highest weights of a representation

Recall that given a representation $\pi$ of $\mathfrak{sl}_n$, a weight $\mu$ is said to be of highest weight if its corresponding weight vector is annihilated by all the positive root spaces (1). ...
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100 views

Three dimensional Lie algebra L with dim L' = 1

Now suppose the derived algebra has dimension 1. Then there exits some non-zero $X_1 \in g$ such that $L' = span\{X_1\}$. Extend this to a basis $\{X_1;X_2;X_3\}$ for g. Then there exist scalars ...
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1answer
136 views

Cartan subalgebra of simple Lie algebra

I could not get the following, could someone give me a hint? Let $\mathfrak{H}$ be a Cartan subalgebra of a simple Lie algebra $\mathfrak{L}$. Show that $\mathfrak{H}$ is abelian. So, we need to ...
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101 views

finding highest weight of dual of a representation of a semisimple lie algebra

If $V$ is an irreducible representation of a semi simple lie algebra having highest weight $\lambda$ then what will be the highest weight of the corresponding irreducible representation $V^*$ (Dual of ...
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62 views

Followup question in Brian Hall's Lie Groups and Algebras.

In ex 9, page 60, he writes down that in order to prove that each invertible matrix $A$ can be written as $A=e^X$, where $X\in M_{n\times n}$, one need to use the fact that if $A$ is unipotent then ...
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2answers
621 views

Example of two-dimensional non-abelian Lie algebra?

can some one give me an example of two-dimensional non-abelian Lie algebra?
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791 views

Two Dimensional Lie Algebra

I read in mark wildon book "introduction to lie algebras" "Let F be any field. Up to isomorphism there is a unique two-dimensional nonabelian Lie algebra over F. This Lie algebra has a basis {x, y} ...
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53 views

Basics of Lie 2-algebras?

Could somebody (simply) explain the basics foundations of Lie 2-algebras, and some basic practical applications ? For instance, does it exist a 3-map (equivalent to the 2-map commutator for Lie ...