Questions tagged [lie-algebras]
For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.
3,787 questions
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The action map induces a Lie algebra homomorphism
Let $G$ be a Lie group and $M$ be a manifold such that $G$ acts on $M$ transitively. Let $$\phi: G\times M\to M$$
be the action map. Now $\phi $ induces a homomorphism
$$\alpha:G\to Aut(M)$$
How to ...
0
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0answers
15 views
Problem with universal enveloping algebra generated by a single element (Jacobson)
Jacobson's book on "Lie algebras" has the following definition of enveloping algebra generated by a subset (Definition 2, Chap II) : Start with an unital associative algebra $A$ (over a field $F$) and ...
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1answer
30 views
Lie Bracket and Matrix Groups
The lie bracket appears in manifolds and matrix groups. For manifolds a tangent vector $X$ is $$X(p)=\sum_{i=1}^n a_i(p) \frac{\partial}{\partial x_i}$$ where there is the parametrization $\mathbf{x}:...
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1answer
38 views
Converse of Schur's lemma for Lie algebras. [duplicate]
The statement of Schur's lema for Lie algebras says that.
Let $(\rho,\mathcal{V})$ is a complex irreducible finite-dimensional representation of a Lie algebra $\mathfrak{g}$. If $T$ conmmutes with $\...
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0answers
29 views
Deriving Lie Algebra for a Lie Group
I'm reading Helgason's DG, Lie Groups, and Symmetric Spaces and at one point he briefly mentions the Lobatchevski half-plane on page 136:
The group $G$ of the mappings $T_{a,b}: x \to ax+b$ with $x ...
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1answer
13 views
Irreducibility of the derived representation
Let $G$ a linear Lie group with lie algebra $\mathfrak{g}$.
If $(\pi, \mathcal{H})$ is a irreducible representation of $G$. Does the irreducibility of $\pi$ imply the irreducibility of derived ...
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1answer
82 views
Connected linear Lie groups $G$ generated by $\exp (\mathfrak{g})$.
Let $G$ be a connected linear Lie group with Lie algebra $\mathfrak{g}$.
I understand that any open neighborhood of the identity of $G$ generates it, but, why does $\exp(\mathfrak{g})$ also generate ...
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0answers
27 views
Degree of universal cover of simple Lie group
I have seen a statement that if $\mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $\mathfrak{g}$. Equivalently, the simply connected group with Lie ...
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1answer
39 views
How to show that Symplectic group $Sp(n,\mathbb C)$ is not compact?
I wanted to prove above just using basic facts .I had only given that Sympletic group preserve following bilinear form :
$B[x,y]=\sum_{i=1}^kx_iy_{n+i}-x_{n+i}y_i$
I had defination of compactness as ...
2
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0answers
10 views
Bilinear form on tensor product restricted to direct summands
Let $\mathfrak{g}$ be a complex semi simple Lie algebra. Then $\mathfrak{g}$ is equipped with a canonical $\mathfrak{g}$-invariant non-degenerate bilinear form $\beta$. Now this gives a $\mathfrak{g}$-...
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0answers
7 views
Lie group map whose differential is an isomorphism is a covering map
While trying to read the proof in Fulton and Harris of their “Second Principle,” I ran across something that I do not understand. They seem to claim that if $f: G\rightarrow H$ is a map of Lie groups ...
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0answers
21 views
Adjoint functors for Lie Algebras
Let's restrict to finite dimensional case.
Functor $(-)^{\mathrm{ab}}: \mathrm{LieAlg} \to \mathrm{AbLieAlg}$ is left adjoint of the inclusion functor $i: \mathrm{AbLieAlg} \to \mathrm{LieAlg}$. ...
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0answers
27 views
Differential form ''equation''
I am having a bit of a trouble with the following. I'm working in the homogeneous Lie Group $\mathbb{R}\ltimes \mathbb{R}^3$ with an specific bracket an it give me de following system to integrate and ...
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1answer
38 views
Connection between a Lie algebra's root system and it's Lie bracket
I have for some time been trying to understand what the root systems of Lie algebras "mean". I understand that vaguely speaking, the Lie algebra is the derivative of the corresponding Lie group at the ...
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1answer
28 views
Determinant of adjoint representation
Let $G$ be a semisimple Lie group with Iwasawa decomposition $G=KAN$ and consider the determinant of the adjoint representation $\operatorname{Ad}$ of $AN$.
I want to determine what the derived ...
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1answer
87 views
Existence of a semisimple elliptic subalgebra
Let $\mathfrak{g}$ be the Lie algebra of a compact Lie group $G$. Denote by $\mathbb{C} \mathfrak g = \mathbb{C} \otimes \mathfrak g$ the complexification of $\mathfrak g$. If necessary, I would not ...
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2answers
36 views
Definition of the Lie algebra and the Lie bracket for general vector fields
I've started to go deep into the theory of Lie groups to eventually understand their representation theory. I picked up a text online and right on the first chapter something started to bother me. ...
2
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1answer
51 views
Existence of a non essentially real ideal in a semisimple Lie algebra
Let $\mathfrak{g}$ be the Lie algebra of a semisimple compact Lie group $G$. Denote by $\mathbb{C} \mathfrak g = \mathbb{C} \otimes \mathfrak g$ the complexification of $\mathfrak g$. I am assuming ...
3
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0answers
18 views
Decomposing $k$th exterior powers $\Lambda^kV(\omega_1)$
Let $\Phi$ be a $G_2$ root system, $\omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V
= V(\omega_1)$ of highest weight $\...
2
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1answer
28 views
Unique faithful $7$-dimensional representation of semisimple Lie Algebra with $G_2$ root system
I am asked to show that if $\mathfrak{g}$ is a semisimple Lie Algebra with root system of type $G_2$, then it has a unique, $7$-dimensional faithful representation.
To start, let $\omega_1, \omega_2$...
6
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2answers
413 views
If two rotation matrices commute, do their infinitesimal generators commute too?
Suppose that $e^A$ and $e^B$ are two rotations in $\mathrm{SO}(n)$. If $e^{A}e^{B} = e^{B}e^{A}$, can we conclude that $e^{A+B}=e^Ae^B$? More importantly, can we say that $AB=BA$?
I'm particularly ...
3
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1answer
35 views
Highest Weights of Defining and Adjoint Representations of $\mathfrak{so}_5$
I am asked to describe the defining representation of $\mathfrak{sp}_4$ in terms of highest weights, and then I am asked to repeat this process for the defining and adjoint representations of $\...
1
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1answer
38 views
Understanding proof of PBW 1
This is first of series of questions I have in understanding PBW. I am following this note. Ring $R$ is assumed to be commutative.
Definition 1: Let $\mathfrak{g}$ be a Lie algebra over a ...
4
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1answer
49 views
Irreducible Dual Representation
For a semisimple Lie Algebra $\mathfrak{g}$ with Cartan Subalgebra $\mathfrak{t}$, let $V(\lambda)$ be the unique irreducible highest weight module with highest weight $\lambda$.
I am asked to show ...
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1answer
16 views
Calculating the Formal character on the irreducible $(n+1)$ dimensional representation of $\mathfrak{sl}_2$
Let $V(n)$ be the unique, irreducible representation of $\mathfrak{sl}_2$ of $(n+1)$-dimensions.
Let $\rho$ be the sum of all fundamental weights.
I want to calculate the formal character $ch(V(n)) ...
2
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1answer
34 views
The Character/Weight of a Representation of an Algebra
This may well be something of a silly question, but if so, then all the more reason I get it straightened out.
I have in the past been working with representations of both groups and algebras, and in ...
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0answers
65 views
$SU(2)$ and its representations
Does the Lie-Algebra of $SU(2)$ always have 3 generators? Because I'm reading about different representations of $SU(2)$ as $2\times2$, $3\times3$, $4\times4$ matrices etc. But I guess that even if we ...
2
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1answer
28 views
Subgraphs of Dynkin Diagrams
Am I right in thinking that if we have two semisimple Lie Algebras $\mathfrak{g} $ and $\mathfrak{h}$ with respective Dynkin Diagrams $A$ and $B$, we may find an injective homomorphism of Lie Algebras ...
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0answers
33 views
Which roots are fixed by simple reflections of the Weyl Group?
Let $\Phi$ be a root system of a semisimple Lie Algebra, and $W$ it's Weyl group. Let $\Delta = \{ \alpha_1, \dots, \alpha_l \}$ be a root basis, and let $w_i \in W$ be the simple reflection ...
2
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1answer
36 views
Is this pseudo-Cartan decomposition of $SO(n)$ valid?
I'm a graduate student in a field of science where we frequently need to optimize a matrix in $U(n)$ or $SO(n)$ (henceforth $SO(n)$ for concreteness) to get an "optimal" orthonormal basis before doing ...
0
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1answer
36 views
Lie algebra generator that does not appear on rhs of the algebra
Is it possible that in a lie algebra one may have a generator that does not appear on any of the commutators?
If the above line is too physicisty for you, maybe I can try to translate to a more ...
1
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1answer
31 views
$\frak{g}^{\bot}$ $=Z(\frak{g})$ if and only if $\frak{g}$ is reductive
Let $\frak{g}$ be a finite dimensional Lie algebra over $k$, a field of characteristic $0$.
Recall that $\frak{g}$ is called reductive if the center $Z(\frak{g})$ is equal to the radical, rad$(\frak{...
2
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1answer
39 views
The intersection of a maximal toral subalgebra with a simple ideal of a Lie algebra is a maximal toral subalgebra of the simple ideal.
I'm reading Humphreys' Introduction to Lie Algebras and Representation Theory and I have a question about Corollary 14.1, which reads:
Humphreys Corollary 14.1. Let $L$ be a semisimple Lie algebra,...
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1answer
18 views
Prove that $[\frak{g}$ $,cen(I)] \neq cen(I)$
Let $\frak{g}$ be a finite dimensional Lie algebra over an algebraically closed field, $k$, of characteristic $0$.
Suppose that $I \triangleleft \frak{g}$ is an ideal of co-dimension $1$, and that $\{...
4
votes
1answer
45 views
Is there a relationship between pre-Lie algebras and post-Lie algebra?
You can find a short survey in this paper https://arxiv.org/pdf/1712.09415.pdf on Post-Lie algebras. I am interested in them because both of them can be constructed on rooted trees. But I don't know ''...
2
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0answers
24 views
Classification of metric Lie Algebras of dimension 7?
I want to classify all metric two-step Lie algebras of dimension 7. Clearly, the dimension of the center of such Lie algebras is $\leq 5$. So, I have to consider 5 cases separately, where the ...
0
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1answer
46 views
The lattice generated by $\{w(\rho) - \rho\,\vert\,w\in W\}$
Consider an irreducible root system associated to a complex simple Lie algebra $\mathfrak{g}$. Let $\rho$ be the half sum of positive roots and let $W$ be the Weyl group. Then what is the lattice $L$ ...
7
votes
1answer
60 views
Derived series of Lie algebra in reverse
Consider the first element in the derived series of a Lie algebra $L$,
defined by $L^{(1)}:=[L,L]$. For a given Lie algebra $\tilde{L}$, is
there always a Lie algebra $L$ such that $L^{(1)}=\tilde{L}$...
2
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0answers
33 views
Reference for Jacobson's theorem about lie algebras
My book (Lie groups by Postnikov) has the following theorem, which it calls Jacobson's theorem:
Let $A$ be a unital associative algebra over some field of zero characteristic, $X$ a subset of $A$ ...
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1answer
114 views
Lie algebra of a matrix group
Let $G$ be a Lie group of matrix. I can define two Lie algebras from there :
$G'$: the set of matrices obtained by computing componentwise the derivative $\gamma'(0)$ of every paths $\gamma$ in $G$ ...
0
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1answer
56 views
Effect of a Lie group morphism associated to a Lie algebra morphism to wedge product
I currently struggeling with the last exercise on my assignment:
Fix $\omega\in\bigwedge^3(\mathbb R^n)^*$. Let $G$ be a Lie group, $\rho\colon G\to GL(n,\mathbb R)$ a Lie group morphism such that
$$
...
9
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1answer
171 views
Modifying unitary matrix eigenvalues by right multiplication by orthogonal matrix
I have a matrix $U \in U(n)$ ($U^* U=Id$), with eigenvalues $\lambda_1, \dots \lambda_n \in S^1$. I would like to know if its always possible to find a matrix $O \in O(n)$ such that the eigenvalues $\...
0
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1answer
19 views
Irreducible highest weight representations as a graded algebra
Let $L$ be a semisimple Lie algebra and let $V(\lambda)$ be a finite dimensional irreducible $L$-module with the highest weight $\lambda$. How can we view the sum
\begin{align*}
\oplus_{n\in\mathbb{N}...
1
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1answer
16 views
Linear map connecting two left-invariant one-forms valued in different Lie algebras
How to see that a left-invariant one-form on a Lie group valued in a different Lie algebra can be factorized through the canonical left-invariant Maurer-Cartan form of this Lie group followed by a ...
3
votes
0answers
20 views
Is the vertex algebra associated to a negative definite even lattice a vertex operator algebra?
Is the lattice vertex algebra associated to a negative definite even lattice a vertex operator algebra? In particular if $L=\mathbb{Z}\alpha$ with $<\alpha, \alpha>=-2$, does $V_L=\bigoplus_{n \...
5
votes
1answer
121 views
If $(e^{\alpha(t)})'=\alpha'(t)e^{\alpha(t)}$ then $\alpha(t), \alpha'(t)$ commute?
Let $\alpha(t)$ be a smooth path of real $n \times n$ matrices. (Formally $\alpha:(-\epsilon,\epsilon) \to M_n(\mathbb{R})$).
If $(e^{\alpha(t)})'=\alpha'(t)e^{\alpha(t)}$ for every $t$ or $(e^{\...
0
votes
1answer
13 views
One dimensional image of the adjoint action
Let $K$ be an algebraically closed field of characteristic zero, and let $\mathfrak{g}$ be a finite dimensional, nilpotent Lie algebra over $K$.
My question is, can we find an element $x\in\mathfrak{...
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2answers
25 views
Normal subgroup in a matrix Lie group
Prop: If $G$ is a Matrix Lie group, then the connected component that contains the identity $I$ is a normal subgroup of $G$.
I have problem in the proof of this. Suppose that $A$ and $B$ belong to ...
1
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1answer
38 views
Lie algebra generator relation $T^a T^b \propto T^c $ valid for any $a,b$?
Given a Lie Algebra (such as $su(n), so(n))$ can I always find a set of generators + identity $\{T^a\}\cup \{id\}$ such that there exists a $c$ for any given $a,b$ such that $T^a T^b = C(a,b) T^c $ ...
0
votes
2answers
35 views
Killing form for a different representation
In this Math Overflow question, the OP was asking if the Killing form defined for a different representation (than the adjoint) was related to the normal Killing form (defined w.r.t. the adjoint rep.):...
