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.
1
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0answers
25 views
Derivations of $\frak{gl}_n(\mathbb{C})$
Let $\frak{g} = \frak{gl}_n(\mathbb{C})$.
What is the Lie algebra of derivations $\text{Der}(\frak{g})$?
Recall a Lie algebra derivation is a linear map $f: \frak{g} \to \frak{g}$ such that $f([x,...
1
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0answers
19 views
Lévy processes, Brownian motion and Lie groups
We know that Brownian motion describes any (non-deterministic) Lévy process with continuous sample paths.
The above statement is true in Euclidean space. My question is: does it stand for other ...
3
votes
1answer
50 views
If $\mathfrak{h}$ is a maximal ideal, then there exists a subalgebra $\mathfrak{l}$, such that $\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{l}$
Definition:
Let $\mathfrak{g}$ be a Lie algebra, a ideal $\mathfrak h \subset \mathfrak g$ is called a maximal ideal, if for all ideal $\mathfrak{h}_1 \subset \mathfrak{g}$ such that $\mathfrak{h}\...
1
vote
1answer
43 views
Nilpotence criterion for solvable Lie algebras
Let $\mathfrak{g}$ be solvable Lie algebra. Lie’s theorem states, that adjoint representation is a homomorphism $\operatorname{ad}:\mathfrak{g}\to \mathfrak{t}$, where $\mathfrak{t}$ is an algebra of ...
1
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0answers
25 views
Subgroups of solvable Lie groups are closed
I was reading this article about solvable Lie groups. I quote:
"An arbitrary connected subgroup of a simply-connected solvable Lie group is closed and simply connected"
He refers this to a paper ...
1
vote
1answer
34 views
Showing the injective map from $\mathfrak{g}$ to $U(\mathfrak{g})$.
From Theorem 9.7 in Lie groups, Lie algebras, and representations written by Hall,
the linear map $ i : \mathfrak{g} \rightarrow U(\mathfrak{g})$ is defined by
For all $X$, $Y \in \mathfrak{g}$, $i(...
2
votes
0answers
32 views
Improve Lie algebra structure constant formula $f^{abc} f^{ade} \propto \delta^{b,d}\delta^{c,e}-\delta^{b,e}\delta^{c,d} + …?$
This is really a simple naive question.
We know Levi-Civita symbol $\epsilon^{abc}$ has a nice property: https://en.wikipedia.org/wiki/Levi-Civita_symbol#Proofs
$$
\epsilon^{abc} \epsilon^{ade}=\...
0
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1answer
21 views
Definition of Hamiltonian Action from Ana Cannas da Silva's book
I've been learning about Lie Groups, Lie Algebras and moment maps and my aim is to get to Sympletic Reduction. Currently I'm struggling with a few details of the definition of hamiltonian action from ...
0
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0answers
14 views
Iwasawa Decomposition of $Gsp_n$
My question is how to find the Iwasawa Decomposition of $Gsp_n=\{g\in GL_{2n}:g^tJg=\mu(g)J, \mu(g) \mathrm{is\ a\ constant}\}$, where $J=\left(\begin{array}{cc}
& I_n \\
-I_n &
\end{...
1
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0answers
39 views
Determining the connected subgroups of a matrix (Lie) group
Let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{gl}(n, \mathbb{R})$ and $\exp(\mathfrak{g})$ its image by the exponential map. Although this set itself might not be a group, its (algebraic) ...
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0answers
16 views
Inhomogeneous Lie derivative equation on a Lie group
Let $G$ be a connected Lie group and let $\xi_i$, $i=1,...,n$ be a basis of its Lie algebra (say, of left-invariant vector fields). We let $B_i$, $i=1,...,n$ be given symmetric sections of $TG \otimes ...
0
votes
0answers
14 views
Relation between central and derived series in Lie algebras and Lie groups
I wonder if there exists a one-to-one correspondence between the terms of the derived series of a Lie algebra and a Lie group. In that case, I suppose that the Lie algebra associated to one term in ...
0
votes
2answers
39 views
Group action on convex cone
I was wondering if I could get some help understanding the following fact in an academic paper. The setup is as follows:
An open subset of $\Omega \subset \mathbb R^k$ is an open convex cone if it ...
2
votes
0answers
33 views
Jacobson-Morozov Theorem to Produce a One-Parameter Subgroup
Let $X\in\frak{sl}_2(\Bbb{R})$ be nilpotent. Prove that there is a one parameter subgroup $\lambda:\Bbb{R}^\times\rightarrow\mathrm{SL}_2(\Bbb{R})$ such that $\mathrm{Ad}(\lambda(t))\cdot X = t^2X$ ...
1
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0answers
12 views
$(\mathfrak{g},K)$-Module where $K$ is a field?
I am reading through Finite groups of Lie type_ conjugacy classes and complex characters by Roger Carter, and came across this passage where Carter is setting up a special class of module to give a ...
1
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0answers
16 views
Fundamental, Regular, and Defining representations
Are these three representations for compact Lie groups and its Lie algebra mean the same thing? Let's focus on "classical" cases like $SO(N)$ and $\mathfrak{so}(n)$ for concreteness (unless I need to ...
0
votes
1answer
24 views
Campbell-Baker-Hausdorff Theorem Proof from Stillwell
I am currently working through the proof of the Campbell-Baker-Hausdorff theorem in Stillwell's Naive Lie Theory. He starts with letting
$$e^Ae^B=e^Z, \qquad Z=\sum_{i=1}^\infty F_i(A,B)\qquad (*)$$
...
0
votes
1answer
26 views
centralizers of nilpotent element in simple Lie algebra and associated Levi subalgebra
Let $\{e,h,f\}$ be a $sl_2$ triples in simple Lie algebra $\mathfrak g$ with usual relations $[h,e]=2e,~ [h,f]=-2f,~[e,f]=h$. Then the centralizer of $e$ is $\mathfrak g_e=\{b:[b,e]=0\}$ and the ...
0
votes
1answer
67 views
generators of Sp(8)
The group $Sp(8)$ has 36 generators which obey the relation
$J=\begin{pmatrix}
0 & I_{4}\\
-I_{4} & 0\\
\end{pmatrix}$
such that $G^TJG=J$ where $G$ are the generators of the group.
I would ...
0
votes
0answers
52 views
Carter's construction of the Chevalley group (Case $A_1$)
I am currently working through Carter's book "Simple Groups of Lie Type", trying to explicitly construct the generators of the Chevalley group of the root system $A_1$.
In the context of $A_1$ let $\...
1
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0answers
23 views
Haar-measure in KAK decomposition of $SL(2,\mathbb{R})$
I would like to know if there is any way to calculate the Haar-measure of the Lie group $ SL (2, \mathbb{R}) $ with respect to the $ KAK $-decomposition, where:
for each $ g \in SL (2, \mathbb{R}) $, ...
0
votes
1answer
24 views
A direc sum of nilpotent Lie algebras such that $C^n(\mathfrak{g}) \subsetneq \mathfrak{z}(\mathfrak{g})$
It is known that the center of nilpotent Lie algebra is never trivial as it always contain $C^n(\mathfrak{g})$ if $\mathfrak{g}$ is nilpotent of class $n+1$
Let $C^n$ denote descending central series....
0
votes
1answer
23 views
morphism $df : Lie(\Bbb R, +) \to Lie(S^1)$
Consider a morphism $f : (\Bbb R, +) \to \Bbb S^1$.
Prove $df : \Bbb R → Lie(\Bbb S^1 ) = i\Bbb R$ is of the form $df : t → 2πiαt, α ∈ \Bbb R$.
My attempt:
$df : Lie(\Bbb R)=\Bbb R → Lie(\Bbb S^1)= ...
1
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0answers
27 views
Root system of Lie algebra: right angle => not simple? [duplicate]
Why is it that if the roots of a Lie algebra form angles of 90 degrees, the Lie algebra is not simple? Is it because the said roots commute with each other and so the Lie algebra can be broken down to ...
0
votes
0answers
29 views
killing Form is null for some elements of Lie algebra of symplectic group
Let $\mathfrak{g} = \{X ∈ gl_{2n} (\Bbb R), ^tXJ + JX = 0\}$ where $J=\begin{pmatrix}
0 & I_n \\
-I_n & 0
\end{pmatrix}$
Let $i ∈ [1, ...,n]$ and $H_i = E_{i,i} − E_{n+i,n+i} (i = 1, ..., n)$....
5
votes
1answer
39 views
Does usual Jordan decomposition be preserved by finite dimensional representation?
If $\mathfrak g\subseteq \mathfrak{gl} (V)$ is a finite dimensional Lie algebra, then we have a usual Jordan decomposition in $\mathfrak{gl} (V)$,
$$
\forall x \in \mathfrak{g}\ \exists x_s, x_n\in \...
0
votes
1answer
20 views
Definition of reductive homogeneous space
I'm studying homogeneous spaces from the book "An Introduction to Lie Groups and the Geometry of Homogeneous Spaces" by A. Arvanitoyeorgos.
I have some problems understanding the definition of ...
0
votes
1answer
36 views
Show invariance under linear transformation [closed]
A simple question but I'm currently stuck.
Let $\kappa\in\mathcal{R}$, and let $\sigma = \kappa I$ and $\pi$ be real $2\times2$ matrices, where $I$ is the $2\times2$ unit matrix. Define the map $R: (\...
0
votes
0answers
18 views
Lie Group and Lie Algebra Homomorphisms
I am reading Brian C Hall's Lie Groups, Lie Algebras, and Representations and trying to do one of the exercises about showing that isomorphic matrix Lie groups give rise to isomorphic Lie Algebras. ...
1
vote
1answer
18 views
How can we define a $\mathbb{Z}$-grading on sl2
Let $sl_{2}$ be $2 \times 2$ traceless matrices over field $K$ of characteristic $0$. How can we define a $\mathbb{Z}$-grading on $sl_{2}$?
Let consider $h ,e ,f$ as follows respectively:
\begin{...
0
votes
0answers
24 views
About Levi factor of a standard parabolic subalgebra
Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Fix a Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$ and a parabolic subalgebra $\mathfrak{p}$ ...
0
votes
1answer
21 views
About root space decomposition of complex semisimple Lie algebra
It is well-known that for a complex semisimple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$ and root system $\Phi$, there is a root space decomposition
$\mathfrak{g}=\mathfrak{h}\...
0
votes
1answer
29 views
Dimension of the derived algebra $L'$ of a 3-dimensional Lie algebra over a field $\Bbb{F}$.
From Erdmann and Wildon's Introduction to Lie algebras.
If $L$ is a non-abelian 3-dimensional Lie algebra over a field $F$, then we know only that the derived algebra $L'$ is non-zero. It might ...
2
votes
2answers
47 views
Lie group $SU(2)$ is the universal covering group of $SO(3)$.
I need to show that Lie group $SU(2)$ is the universal covering group of $SO(3)$ using the Adjoint representation of $SU(2)$.
But I am stuck at the first step of finding the adjoint representation of ...
0
votes
0answers
12 views
Structure constants of two isomorphic Lie algebras.
Proposition: Let $L_1$ and $L_2$ be Lie algebras. Then $L_1$ is isomorphic to $L_2$ if and only if there exist a basis $B_1$ of $L_1$ and $B_2$ of $L_2$ such that the structure constants of $L$ with ...
1
vote
1answer
29 views
Lie algebra of a closed discrete subgroup is zero.
If $H$ is a closed subgroup of Lie group $G$, then show that $\mathfrak{h}=0$ if and only if $H$ is discrete, where $\mathfrak{h}$ is the Lie algebra of $H$.
We know that $\mathfrak{h}=\{X\in \...
2
votes
1answer
33 views
Does a basis for a Lie algebra of a Lie group $G$ depend on whether $G$ is embedded?
My professor recently gave the problem to find a basis for a lie algebra of a given embedded lie subgroup. The problem stressed that the lie subgroup was embedded (which was clear from the definition ...
0
votes
1answer
40 views
Derivative of products of exponential maps
Let $G$ be a (finite-dimensional) Lie group with Lie algebra $\mathfrak g$. Then for $f\in C^\infty(G)$, $X,Y\in\mathfrak g$ and $g\in G$ one can define $f(ge^{tX})\in C^\infty(\mathbb R)$ which ...
1
vote
1answer
37 views
Schur's Lemma in Infinite Dimensional Lie Algebras
Let $\mathfrak{g}$ be a $\mathsf{k}$-Lie algebra, with $(\rho, V)$, $(\sigma, W)$ irreducible $\mathfrak{g}$-representations. Then the easy part of Schur's Lemma states that a $\mathfrak{g}$-linear ...
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votes
0answers
9 views
number of connected components of $E(n)$
Let $E(n) := \{f : \Bbb R^n → \Bbb R^n , ||f (x)|| = ||x||\}$ be the group of affine isometries of $\Bbb R^n$.
Prove that $T(n)$ the set of translations $x→ x + y, y ∈ \Bbb R^n$ verifies $T(n) \...
2
votes
2answers
24 views
Definition of trace of an adjoint representation.
What does the trace of an adjoint representation mean? I was asked to prove the following
If $z \in L'$, then $\operatorname{tr}(\operatorname{ad}z)=0$.
I know what a trace of a matrix is, but ...
0
votes
1answer
25 views
$ (H, \alpha) $ is lie subgroup of $G$, $\exp(tY) \in \alpha(H)$, How to prove $Y \in \alpha_\ast (h)$?
$ (H, \alpha) $ is lie subgroup of $G$, $\exp(tY) \in \alpha(H)$, $\forall t \in \mathbb{R}$, How to prove $ Y \in \alpha_\ast (h)$? where $h$ is the lie algebra of $H$.
When I encounter this exc, ...
1
vote
1answer
31 views
Subgroups of semisimple Lie groups
Let $G$ be a connected semisimple Lie group.
If $N$ is a normal closed semisimple Lie subgroup of $G$ and $\Lambda$ is a closed discrete subgroup of $G$. Is $N\Lambda$ a closed subgroup of $G$?
If $...
3
votes
2answers
60 views
How can a velocity vector be determined literally as a derivative in the calculus sense?
I am following Tu's book on manifolds. On pages 178-179 he proves that the Lie algebra of $\mathrm{SL}(n,\mathbb{R})$ is the set of traceless real $n\times n$ matrices. Specifically he writes:
...
1
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0answers
29 views
Lie algebra homomorphism $\varphi: \frak{g} \to h$ not of the form $D_0\phi$
I'm looking for two connected Lie groups $G, H$ with Lie algebras $\frak{g}, h$ and a Lie algebra homomorphism $\varphi: \frak{g} \to h$ that isn't of the form $D_0\phi$ for any Lie group homomorphism ...
1
vote
1answer
29 views
Lie algebra homomorphisms preserve semi-simplicity?
Is the following proof correct?
Claim: Let $g$ be a semi-simple Lie algebra, and $f: g\rightarrow h$ be a homomorphism of Lie algebras. Then $Im f \leq h$ is a semi-simple Lie algebra.
Proof ...
1
vote
1answer
59 views
Maximal nilpotent subalgebra which is not a Cartan Subalgebra
Consider the special linear algebra $\mathfrak{sl}_2(\mathbb{F}$) and let $\mathfrak{n}_2 \subset \mathfrak{sl}_2(\mathbb{F})$ be the subalgebra of strictly upper triangular matrices. Clearly $\...
0
votes
0answers
39 views
Making the algebraic similarities of groups and Lie algebras precise.
There is a correspondence between Lie algebras and groups on the level of their "algebra", in that many "purely algebraic" theorems in group theory correspond exactly to "purely algebraic" theorems in ...
2
votes
1answer
35 views
Why is the Chevalley-Eilenberg differential a coderivation?
For every Lie algebra $\mathfrak{g}$ we can consider the Chevalley-Eilenberg complex given by the exterior powers $\bigwedge^n \mathfrak{g}$ together with the differential $d_{\mathrm{CE}} \colon \...
2
votes
0answers
30 views
Homology of solvable Lie algebras
Let $\mathfrak{g}$ be a solvable lie algebra and $\lambda\in (\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be the character of $\mathfrak{g}$. How to compute homology for $\mathbb{C}_\lambda$, the ...
