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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.

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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,...
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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 ...
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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}\...
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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 ...
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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 ...
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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(...
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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}=\...
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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 ...
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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{...
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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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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 ...
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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 ...
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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 ...
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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 $\mathrm{Ad}(\lambda(t))\cdot X = t^2X$ ...
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$(\mathfrak{g},K)$-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 ...
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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 ...
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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 $$e^Ae^B=e^Z, \qquad Z=\sum_{i=1}^\infty F_i(A,B)\qquad (*)$$ ...
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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 ...
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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 ...
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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 $\...
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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}) $, ...
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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....
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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)= ...
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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 ...
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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 $H_i = E_{i,i} − E_{n+i,n+i} (i = 1, ..., n)$....
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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 \...
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1answer
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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 ...
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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: (\...
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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. ...
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1answer
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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{...
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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 $\mathfrak{p}$ ...
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1answer
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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}\...
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1answer
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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 ...
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2answers
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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 ...
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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 ...
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1answer
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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 \...
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1answer
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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 ...
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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 ...
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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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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) \...
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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 ...
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1answer
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$ (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, ...
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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 $...
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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: ...
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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 ...
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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 ...
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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 $\...
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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 ...
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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 \...
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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 ...