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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Why the induced metric from the lie algebra of lie group $G$ is left invariant.

we say that a Riemannian metric on $G$ is left invariant if $<u,v>_y = <d(L_x)_y u,d(L_x)_y v>_{L_x(y)}$ to introduce a metric on $G$, take any arbitrary inner product $< , >_e$ on ...
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1answer
32 views

Derived algebras and solvable Lie algebras

The idea of a Solvable Lie algebra hinges on the definition of the sequence: $$ g \ge [g,g] \ge [[g,g],[g,g]] \ge [ [[g,g],[g,g]] , [[g,g],[g,g]] ] \ge \ldots $$ and its limiting group. If an ...
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28 views

Limit of the commutator of two elements?

Given a Lie group $G$ such the $\mathfrak{g}$ denoted its Lie algebra. Let $[g,g']_{G}$ the commutator of two elements $g,g' \in G$ and denoted by $[X,X']_{\mathfrak{g}}$ the Lie bracket of two ...
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1answer
22 views

Where I can find classifications of Lie algebra $A_{4.7}^{-1}$?

My Lie algebra with commutation relation $[e_2, e_3] = e_3,\;[e_2, e_4] = -e_4,\;[e_3, e_4] = -e_1$ is isomorphic to Lie algebra $A_{4.7}^{-1}$ through transformations $e_1\mapsto e_1,\;e_2\...
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25 views

compact lie group -> real analytic orbits in $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Now, $G$ also acts on $\mathfrak{g}^*$, the dual of the Lie algebra, by the coadjoint-action. My question now is: are ...
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1answer
61 views

Irreducible representations of Heisenberg group

Lately, I've been struggling with the following problem. Let $H$ be the 3 dimensional Heisenberg group and let $\rho:H\to\text{GL}(n,\mathbb{C})$ be a irreducible representation. Show that $n=1$. I ...
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1answer
23 views

what if the infinitesimal generator of a vectorfield vanishes?

Let $(M,g)$ be a riemannian manifold and $H$ a Lie group acting on $M$. Denote by $l \colon H \times M \to M$ and $l_h \colon M \to M$ the action of $H$ on $M$. Now $H$ acts on $TM$ by derivations, ...
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13 views

About exceptional Lie group E6

How to show the group of determinant preserving linear transformations of $ z $ is isomorphic to $$ \{a \in Isom_\mathbb{C}(z^\mathbb{C},z^\mathbb{C})|det(aX)=det(X),<aX,aY>=<X,Y>\} $$ ...
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1answer
74 views

Is the alternating group a lie group [duplicate]

Is the alternating group a lie group. If so what is the lie algebra corresponding to it? This is not a homework questions. I need the dimension of the lie algebra (if one exists) to prove some ...
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21 views

Extension of Lie algebras

Let $L$ be a Lie algebra and $A$ be its subalgebra. Let consider derivation $d: A \to L$. How can we interpret this extension $$H=\langle L,t | [t,a]=d(a), \forall a \in A \rangle? $$ How will the ...
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1answer
47 views

If G is a compact semisimple Lie group, then its Killing form is negative definite

Theorem: If $G$ is a compact semisimple Lie group, then its Killing form is negative definite. In its proof: Since $G$ is compact there is an Ad-invariant inner product on $\mathfrak g$. Since each ...
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1answer
35 views

Radical of a direct sum of Lie algebra

If we take $L$ a finite dimensional Lie algebra on $\mathbb{R}$, $A$ a sub-abgebra and $I$ an ideal of $L$ such that $L=A \oplus I$ as vector spaces. We have that $rad(L)=rad(A) \oplus rad(I)$ as ...
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1answer
13 views

Ideal spanned by monomial of degree $n$ in the Universal Enveloping Algebra

I'm trying to understand the Ado theorem proof which uses the universal enveloping algebra of a Lie algebra. In this proof we use the ideal spanned by all monomial of degree $n$ in the universal ...
2
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60 views

Involutions and Representation of Lie Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$. Is easy to see that ...
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28 views

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$?

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$? This question is inspired by noting that if we have a Hamiltonian Lie group action $G\curvearrowright (M,\...
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1answer
61 views

Cartan decomposition of SO(2n)

I am trying to understand the Cartan decomposition theory, on the following example : $G=SO(2n)$, and $K=U(n)$, and I'm interested in the manifold $G/K$ (an hermitian symmetric space). 1) How do we ...
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0answers
28 views

By which the Heisenberg group is introduced? [closed]

I want to know who was the first who introduced the Heisenberg group and in what year. In the Wikipedia there is just an indication that this group was named in honor of the famous German physicist ...
2
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1answer
48 views

Showing a rep of $sl(2,\mathbb{K})$ is irreducible

Let $V$ be a $m+1$-dim $K$-vector space with char$K=0$. Let $(v_0,v_1,\dots,v_m)$ be a basis of $V(m)$. Now suppose I construct a representation of $sl(2,K)$ on this representation. How do I show ...
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2answers
42 views

The Lie Algebra of $O(n)$ is the set of $n \times n$ skew-symmetric matrices

I'm trying to show that the Lie Algebra for $O(n)$ is the set of $n \times n$ skew-symmetric matrices. Here is what I have so far. Since $O(n)$ is the union of two disjoint subsets, the matrices with ...
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10 views

Graph properties of Bruhat order for the general linear Lie algebra $\mathfrak{gl}$ on $\mathbb{Z}^n$

Let $P = \oplus_{i\in \mathbb{Z}}\mathbb{Z}\epsilon_i$ the free abelian group of infinite rank. Then we have a natural partial order $\leq'$ on $P$, that is, $a \leq' b $ if and only if $b \in a+\sum_{...
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1answer
51 views

Regular elements of a Lie algebra

I'm currently trying to learn about regular elements of a Lie algebra but i'm finding the definition quite abstract and can't seem to find many examples anywhere. One thing i'm really unsure about ...
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1answer
49 views

Representation of of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$

Certain part in my textbook implies that a representation of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$, where $S \in \mathbb Z$, is possible. I am trying to find a path that leads to this ...
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32 views

The linear Lie algebra of a closed linear group is closed

I was reading Knapp's Lie groups beyond an introduction and, in the first pages, he shows that the set of all tangent vectors to a given closed linear group $G$ at the identity matrix, that is $$\...
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18 views

Relation between integrable representations and highest weight representations.

Let $g$ be a simple Lie algebra and $U_q(g)$ the corresponding quantum group. What are the relation between integrable representations and highest weight representations of $U_q(g)$? Are all highest ...
2
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1answer
75 views

Sandwich rule for Lie algebras

On an infinite dimensional vector space an operator can be onto but not one-to-one (and vice versa). This arises the following question. Let $L_1$ and $L_2$ be Lie algebras (infinite dimensional, over ...
4
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89 views

$E_8$ and theta functions

The root lattice $\Gamma_8$ of the exceptional Lie algebra $E_8$ is an eight-dimensional lattice which consists of lattice points in $\mathbb{R}^8$ which with respect to an orthonormal basis $e_1, \...
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1answer
38 views

How to write quotient algebra for normalizer?

For Lie algebra $\mathfrak{g}=\{e1,e2,e3,e4\}$ with commutations $[e1, e3]=\,e1, [e2, e3]=\,\alpha\,e2$, I have calculated normalizer for sub-algebra $\mathfrak{q}=\,\{e1+e2\}$ as $\text{Nor}_{\...
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1answer
27 views

Splitting and non-splitting extensions in Lie algebras

For Lie algebra $S=\{e_{1}, e_{2}, e_{3}, e_{4}\}$ with non-zero commutations: $[e_{1}, e_{3}]=e_{1}, [e_{2}, e_{3}]=\alpha\, e_{2}$ we have $S=e_{4}\oplus L_{3}$, such that $L_{3}=\{e_{1}, e_{2}, ...
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1answer
30 views

Orthogonal basis of a Cartan of a Lie algebra with respect to Killing form.

I am trying to understand orthogonal basis of a Cartan of a Lie algebra with respect to Killing form. For example, let $g=sl_2 = \text{Span}\{h, E, F\}$. Then a Cartan of $g$ is $\mathfrak{h} = \...
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0answers
8 views

Finding a polytope in the Cartan Subalgebra

The finite Coxeter groups can be realized as symmetry groups of (semi)-regular polytopes. Not all semi-regular polytopes can be realized this way, but all regular polytopes can. Some examples of ...
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1answer
16 views

How subsets are defined in Lie algebra?

Consider four dimensional Lie algebra with non-zero commutations: $[e_{2},e_{3}]=e_{1}, [e_{2}, e_{4}]=e_{2}, [e_{3}, e_{4}]=-e_{3}$ having sub-algebras $S_{1}=\{e_{1}, e_{2}\}, S_{2}=\{e_{1}+e_{2}\}...
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3answers
53 views

If a Lie Algebra is solvable, is the corresponding Lie group solvable in the group theoretic sense?

I just started working with Lie Algebras with a professor. The way we defined them is probably the standard way; treat Lie Algebras as tangent spaces at the identity of the Lie Group. Now, consider ...
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Classification of irreducible (g,K)-modules for other g than sl2

Harish Chandra showed how to associate to an admissible representation $(\pi,V)$ of a real semisimple Lie group $G$ the so-called Harish-Chandra module $V_K$ of $K$-finite vectors in $V$. This is a $(\...
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Problem on Root Systems

Let $R$ be a root system on a Euclidean space $\mathbb{E}$ with simple roots $\{\alpha_i : i=1,....,l\}$. If $\alpha$ = $\sum_{i=1}^{l}c_i\alpha_i$ be a root , then show that $\frac {c_i(\alpha_i|\...
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1answer
70 views

How many three and four dimensional Lie algebras are there?

Patera and Winternitz have carried out extensive classification of three and four dimensional Lie algebras. When I tried to look for classification for three dimensional Lie algebra with non-zero ...
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2answers
32 views

if $g$ is a lie algebra what is $g^*$?

Iam trying to learn what a coadjoint orbit is but I can't since everywhere I look the definition involves $g^*$.Something that I googled and didn't find anything. I am not even sure what $g^*$. Is it ...
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1answer
40 views

Left and right invariant vector fields

I'm following Woodhouse's book on geometric quantization and I'm stuck with this problem. Let $R_A$ and $L_A$ be right and left invariant vector fields such that $R_A(e)=A=L_A(e)$, where $A$ is an ...
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0answers
44 views

Correspondence between $\mathfrak{sl}(n,\mathbb{C})$ and $\mathfrak{gl}(n,\mathbb{C})$ representations

There is a one-to-one correspondence between irreducible $\mathfrak{gl}(n,\mathbb{C})-$representations and $\mathbb{C} \times \{ \text{irreducible } \mathfrak{sl}(n,\mathbb{C}) \text{ representations}...
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1answer
44 views

extension of derivation of algebras

I am studying extension of derivations, but I am confused of some notations and maybe symbols! For some more details, I recall a theorem. We have the following theorem : Theorem: Let $A$ be an ...
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1answer
26 views

Does equivariant property commute with matrix exponential?

Given a vector space $V$ and endomorphism $M : V \to V$ we can define the exponential of $M$: $\exp(M) = \sum_{k=0}^{\infty} \frac{1}{k!}M^k$. This has the property that it commutes with conjugation ...
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Laplacian in matrix form?

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
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Does a subalgebra of a Lie algebra $g$ define a Lie subalgebra in the dual $g^*$ if $( g, g^*)$ is a Lie bialgebra?

Question: Let $\mathfrak d = \mathfrak g\bowtie \mathfrak g^*$ be the double of the Lie bialgebra $(\mathfrak g, \mathfrak g^*)$, and let $\mathfrak h$ be a Lie subalgebra of $\mathfrak g$. If $\xi,\...
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For compact Lie groups, is $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$ equivalent to being semisimple?

In Ana Cannas da Silva’s Lectures on Symplectic Geomertry, page 167, semisimplicity is defined in the restricted setting of compact Lie groups by A compact Lie group $G$ is semisimple if $[\...
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2answers
27 views

Computing $\operatorname{ad}_x$ where $\operatorname{ad}$ is the adjoint representation [duplicate]

Let $\operatorname{ad}_x:L\rightarrow GL(L)$ be the adjoint representation. In Humphreys "Introduction the Lie-Algebras and representation theory" one can find this example where $x,y$ and $h$ are ...
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1answer
30 views

References for the representation theory of $SU(2, 1)$

I couldn't find any reference with the representation theory of this specific case. I found some general stuff but never explicit computations or realizations. The only thing I found on $SU(2, 1)$ ...
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41 views

Taylor Series in any vector space?

I am working through Alexander Kirillov, Jr.'s An Introduction to Lie Groups and Lie Algebras, and on page 29 he does something I find puzzling. He claims that, since the exponential map is a local ...
2
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1answer
25 views

Every compact semi-simple Lie group has finite center

I am reading introductory lecture notes on Lie groups and Lie algebras. There it is stated as a fact without proof, that any compact semi-simple Lie group has finite center. Here, semi-simple means, ...
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23 views

Difference between symmetry algebra and symmetry group

What is the difference between symmetry algebra and a symmetry group? I just wanted to know if my understanding is right. Lets say we have a system of differential equations. Then the symmetry group ...
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1answer
34 views

reference request: lie algebra-lie group

I am looking for a reference where I can find a (relatively) elementary and self contained proof of the fact that all real, finite dimensional Lie algebras are the Lie algebra of some Lie group. ...
2
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1answer
23 views

Adjoint orbit of universal covering group

Let ${\frak g}$ be a complex semisimple Lie algebra and $G$ a connected Lie group with Lie algebra ${\frak g}$. Let $\tilde{G}$ be a universal covering group of $G$. Take $X\in{\frak g}$ and consider ...