For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.
8
votes
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204 views
Abelian Cartan subalgebras
If a Lie algebra is semisimple or reductive, its Cartan subalgebras are Abelian, and their elements semisimple.
Are there non-reductive algebras with Abelian Cartan subalgebras all of whose ...
7
votes
0answers
78 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 ...
7
votes
0answers
75 views
Reference for l-adic Lie algebras
I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'.
Is there a standard reference for ...
7
votes
0answers
111 views
How to prove these Lie algebra relations
This is a bit of a basic computational question concerning Lie algebras, but I'm getting kind of bamboozled so I thought I'd post it.
I'm confused about how to perform some computations in Serre's ...
6
votes
0answers
70 views
The center of a simply connected semisimple Lie group
I am learning about Lie groups, and I have the following basic question:
Every Lie group $G$ has a (unique) universal covering group $ \bar G $ that is simply connected, and such that the covering ...
6
votes
0answers
88 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 ...
5
votes
0answers
73 views
Quantum Adjoint Action of the Coordinate Algebra on the Enveloping Algebra
As is well known, any Lie group $G$ has a canonical action on its Lie algebra $\frak{g}$, namely the adjoint action $Ad$. Firstly, let me ask, does this extend to an action of $G$ on its enveloping ...
5
votes
0answers
123 views
Generating function for characters of representations
One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ ...
5
votes
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89 views
Pullback of a 3-form to SU2
I have a left invariant 3-form, $\sigma$ on an simply connected Lie group, $G$ whose value at the identity is $\sigma=\langle[x,y],z\rangle$, where $\langle\cdot,\cdot\rangle$ denotes an invariant ...
5
votes
0answers
109 views
An exercise in Serre's Lie algebra book
Let $k$ be a commutative ring. Prove that a Lie $k$-algebra $\mathfrak{g} = 0$ iff $U\mathfrak{g} = k$. Use the adjoint representaion.
Here is my attempt at it:
The only non-trivial statement is ...
5
votes
0answers
158 views
Root Systems of Lie Groups.
Let $G$ be a compact Lie group assumed to be a subgroup of $U(n)$. Also, let $T$ be a maximal torus of G. Then there exists a basis $\{v_1, \ldots ,v_d\}$ of the Lie algebra of $G$, $\mathfrak{g}$, ...
4
votes
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25 views
Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$
If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
4
votes
0answers
33 views
Exercise in Erdmann's Intro to Lie algebras
I'm working on question 4.8 on page 36 of Erdmann's book called Introduction to Lie Algebras. The question is as follows:
Let $L$ be a Lie algebra over a field $F$, such that $[a,b],b]=0$ for all ...
4
votes
0answers
29 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 ...
4
votes
0answers
69 views
What is good about simple Lie algebras?
Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple.
I wonder what beautiful properties does a ...
4
votes
0answers
121 views
internal direct product of lie groups
If $G$ is a (edit: simply connected)Lie group, when does a direct sum decomposition of its Lie algebra (into a direct sum of subalgebras) correspond to a (semi)direct product decomposition of $G$? ...
4
votes
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88 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 ...
4
votes
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246 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 ...
3
votes
0answers
30 views
adjoint representation completely reducible
Let $\mathcal{A}$ be a Lie algebra. Suppose that adjoint representation of $\mathcal{A}$ is completely reducible (or semisimple). Show that $\mathcal{A}$ can be written as a direct sum of semisimple ...
3
votes
0answers
78 views
Irreducible representations of $\mathfrak{sl}_3\mathbb{C}$
I am working through the exercises in Fulton and Harris's Representation Theory, and am stuck on two on page 189.
Let $\text{Sym}^2V$ denote the second symmetric power of the standard 3-dimensional ...
3
votes
0answers
57 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 ...
3
votes
0answers
57 views
$\Delta \subset \Phi$ is a base in a root system imples $\Delta^\vee \subset \Phi^\vee$ is a base in a root system
(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 ...
3
votes
0answers
44 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 ...
3
votes
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288 views
Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.
I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question.
To start with we work with the $\mathbb{Q}$ version of ...
3
votes
0answers
113 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 ...
3
votes
0answers
82 views
The Nambu bracket
Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $$\{F,G\}_c = \langle\nabla ...
3
votes
0answers
111 views
Integral forms of loop algebras.
The question following is about integral forms for semisimple Lie algebras and loop algebras constructed from them.
Let $\frak g$ a finite-dimensional Lie algebra over $\mathbb C$ and $L(\frak ...
3
votes
0answers
84 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 ...
3
votes
0answers
191 views
Invariant inner product $\langle\,,\rangle$ on a Lie algebra
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We can use the Killing form to identify $\mathfrak{h}$ and $\mathfrak{h}^*$ ...
3
votes
0answers
101 views
On the root systems
Let $\Phi$ be a root system of $E$. $\alpha,\beta\in \Phi$.
Let $\lbrace \beta+i\alpha | i\in \mathbb{Z}\rbrace\cap \Phi$, $\alpha$-string through $\beta$, be $\beta-r\alpha,\ldots,\beta+q\alpha$, ...
2
votes
0answers
16 views
Centralizers of connected linear group and its Lie algebra
If we have that $G$ is a connected linear group and $H<G$ with $\mathfrak{h}$ the lie algebra of $H$ and we define the centralizers of the elements in the following way:
$Z(H):=\{a\in G| ...
2
votes
0answers
23 views
The normalizer of a proper sub-algebra properly contains the sub-algebra in a nilpotent lie algebra.
So I am just making my way in to the theory of Lie Algebras. The question at hand comes from page 14 of Humphreys' "Introduction to Lie Algebra and Representation Theory"
Given a finite dimensional ...
2
votes
0answers
28 views
To what extent are formulas obtained in one Lie group valid in another Lie group with an isomorphic Lie algebra?
In quantum optics, I am trying to explore the group generated by squeezing and rotation operators. These are closely related to area-preserving linear transforms, which they induce on the phase space, ...
2
votes
0answers
29 views
Character of half-spin representation
Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum
$$\sum x_1^{\pm ...
2
votes
0answers
29 views
Radical of $\mathfrak{gl}_n$
I find it intuitive enough that the radical of $\mathfrak{gl}_n\mathbb F$ is the scalar matrices, but I have trouble finding an easy, but complete proof:
Proof. Let $\mathfrak s$ denote the scalar ...
2
votes
0answers
68 views
Lie bracket of vector fields definition equivalence
Lie bracket of vector fields is defined in two ways:
Let $\Phi^X_t$ be the flow associated with the vector field $X$, and let $d$ denote the
tangent map derivative operator. Then
the Lie ...
2
votes
0answers
35 views
Regarding the definition of vector field flow
To make the connection to the Lie derivative, let $t \mapsto \Phi^X_t$
be the 1-parameter group of diffeomorphisms (or flow) generated by the
vector field $ X $. The differential $ d\Phi^X_t ...
2
votes
0answers
22 views
Finding the Lie map
Suppose I have a group homomorphism $\rho:SL(2,\mathbb{C})\to SO_0(3,1)$ given by $\rho(a)X=aXa^*$ and I want to see how the corresponding Lie map $L\rho$ looks like. By definition
$$
...
2
votes
0answers
44 views
The enveloping algebra of a finite dimensional Lie algebra has no zero divisor
Let $L$ be a complex, finite dimensional Lie algebra.
It is well-known that the graded associative algebra of the enveloping algebra $U(L)$ is isomorphic to the symmetric algebra $S(L)$. Therefore ...
2
votes
0answers
16 views
Integer domain enveloping algebra
I must prove that if $L$ is a Lie algebra and denoting $U(L)$ the enveloping algebra, then $U(L)$ hasn't zero divisions (e.g. if $ab=0 \,\,\, a,b \in U(L)$ then $a=0$ or $b=0$). Some ideas?
2
votes
0answers
26 views
When is the Killing form null?
When is the Killing form $\kappa$ of a Lie algebra $\mathfrak g$ null, i.e. $\kappa(\cdot,\cdot)=0$? Surely this is true for any Lie algebra with trivial bracket, but is this the only case? I can't ...
2
votes
0answers
75 views
When is the adjoint representation self-dual?
Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...
2
votes
0answers
48 views
Weyl group of $\mathfrak{sl}(2,\mathbb{C})$
$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{gl}(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the ...
2
votes
0answers
71 views
Commutator formula in infinite dimensions
The commutator formula states that for $A,B$ elements of a Lie algebra,
$$ \lim_{n\to \infty}\left\{ ...
2
votes
0answers
38 views
Are there any infinite dimensional subalgebras of the Witt algebra?
The Lie bracket of elements of the Witt algebra is given by:
$[L_m,L_n]=(m-n)L_{m+n}$
Are there any infinite dimensional subalgebras of the Witt algebra that are not isomorphic to the Witt algebra ...
2
votes
0answers
54 views
Standard parabolic Lie subalgebras and conjugacy
Let $\mathfrak g$ be a given semisimple Lie algebra with corresponding adjoint Lie group $G$. A parabolic subalgebra is any subalgebra containing a Borel subalgebra.
We can pick a Borel ...
2
votes
0answers
45 views
ADE type root lattice
Let $\Phi$ be a root system of ADE type, $L$ is the corresponding root lattice, show that $\Phi=\{\alpha\in L:(\alpha,\alpha)=2\}$, where $(,)$ is the normalized non-degenerate symmetric bilinear form ...
2
votes
0answers
98 views
Decomposing products of spinor representations into anti-symmetric tensors
There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
2
votes
0answers
55 views
Basis of the Engel algebra
If I have a connected, simply connected nilpotent lie group given
by the commutators between the elements of a basis of its Lie algebra
how can I recover the left invariant vector fields?
For ...
2
votes
0answers
40 views
Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.
Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in ...
