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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5
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2answers
157 views

Classification of irreducible representations via Casimirs

Physicists almost always label irreducible representations via Casimirs (e.g., characterizing the irreducible representations of $SO(3)$ by spin). I've been looking far and wide to see the general ...
0
votes
1answer
100 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 ...
1
vote
1answer
35 views

U(m) as a subgroup of SO(2m)

We know $U(m)$ is one the subgroups of $SO(2m)$ acting transitively on the sphere $S^{2m-1}$ (one of the groups in the Borel's list). What is the explicit formula of this embedding (or it's action)?
0
votes
1answer
107 views

Orthogonal subspace relative to the Killing form

I'm following a book in Humphrey's Introdutction to Lie Algebra's and Representation Theory. I'm reading the proof that a semisimple Lie algebra is the direct sum of simple modules. It uses the ...
1
vote
3answers
177 views

Why is it so important for the characteristic value of the field of a lie algebra to not be two for many propositions?

In reading my Lie algebra text, I see a lot of propositions starting with, "If char F does not equal 2, then..." For example, if char F does not equal 2, then o(n,F) is a subalgebra of sl(n,F). I am ...
2
votes
1answer
67 views

Existence of a Lie subgroup

Let $G=SU(k)\times T^1$, $S$ a subgroup of the center of $SU(k)$ ($Z(SU(k)\cong \mathbb{Z}_k$) and $\eta$ a homomorphism from $S$ into $T^1$. Suppose $(S, \eta)$ denotes the subgroup of $G$ contains ...
1
vote
0answers
37 views

Symmetric algebra and complex polynomial on Lie algebra

Let $G$ a Lie group and $\mathfrak{G}$ its Lie algebra. How can I identify the symmetric algebra on $\mathfrak{G}$ ($S(\mathfrak{G})$) with the algebra of complex polynomials on $\mathfrak{G}$, that I ...
4
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0answers
168 views

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 ...
1
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1answer
73 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 ...
2
votes
1answer
186 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 ...
2
votes
1answer
74 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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0answers
52 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 ...
1
vote
2answers
102 views

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

what is the natural action of $\mathfrak{sl}({V})$ in tensor spaces ?
0
votes
1answer
137 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 ...
3
votes
2answers
181 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 ...
2
votes
0answers
69 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. ...
0
votes
1answer
169 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 ...
3
votes
2answers
105 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 ...
2
votes
2answers
95 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 $$\text{ad}: \mathfrak{sl}_2(\mathbb{C})\to\mathfrak{gl}_3(\mathbb{C})$$ ...
1
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0answers
34 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
votes
2answers
263 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 ...
1
vote
1answer
52 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 ...
2
votes
0answers
60 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 ...
3
votes
1answer
122 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 ...
2
votes
2answers
197 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 $\mathfrak g \times M \to M$ ...
1
vote
1answer
60 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
votes
0answers
41 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
votes
1answer
58 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 ...
0
votes
3answers
148 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 $1$. ...
9
votes
0answers
112 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 ...
2
votes
1answer
332 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 ...
1
vote
1answer
41 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 ...
0
votes
2answers
232 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: ...
1
vote
1answer
78 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
votes
1answer
138 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 ...
4
votes
2answers
447 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 $[X,Y] = A$ ...
1
vote
1answer
63 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 ...
0
votes
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 $\mathcal{L}$.
5
votes
1answer
231 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 ...
1
vote
0answers
100 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 ...
4
votes
2answers
194 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 ...
2
votes
0answers
59 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 ...
0
votes
1answer
86 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 ...
3
votes
1answer
96 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: ...
3
votes
0answers
159 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 ...
4
votes
2answers
188 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 ...
0
votes
1answer
54 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 ...
2
votes
1answer
104 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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0answers
66 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? ...
1
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1answer
57 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). ...