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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38 views

Spheres as Homogeneous Spaces

Any odd dimensional sphere $S^{2n+1}$ can be expressed as an homogenous space of $SU(n)$ by $S^{2n+1} \simeq SU(n)/SU(n-1)$. Any even dimensional sphere $S^{2n}$ sphere can be expressed as an ...
2
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1answer
22 views

Right unidimensional extension of Heisenberg Algebra

I'm reading a book on Mathemathical Physics and speaking of the Quantum harmonic oscillator says (this is a translation in english, hope is right): The commutation relations between operators are: ...
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1answer
24 views

Nilpotent 4-dim Lie algebra

What is an example of a nilpotent Lie algebra $\cal g$ with dimension four and: $dim\ \mathcal g'=1$, $dim \ \mathcal g'=2$, $dim \ \mathcal g'=3$? Is there a way to construct them?
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1answer
18 views

3-dim Lie algebra with two commutative elements

Let $\cal g$ be a Lie algebra and let $a,b,c\in \cal g$ be such that $ab=ba$ and $[a,b]=c\not =0$. Let $\mathcal h=span\ \{a,b,c\}$. How to prove that $\mathcal h$ is isomorphic to the strictly upper ...
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1answer
22 views

What role does abstract Jordan decomposition play in the study of semisimple Lie algebras?

I have seen the notion of the abstract Jordan decomposition be given for semisimple Lie algebras over an algebraically closed field of char = 0 in a number of places, a few theorems proved about it ...
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1answer
36 views

The center of a nilpotent Lie algebra intersects each ideal

If $\cal h$ is a nonzero ideal in a nilpotent Lie algebra $\cal g$. How to prove that $\mathcal h\cap Z(\mathcal g)\not =0$, where $Z(\mathcal g)$ is the center of $\mathcal g$?
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1answer
29 views

If L is nilpotent then $K\cap L^n \not=0$

Let $K$ be a proper ideal of a nilpotent Lie algebra $L$. If the nilpotency class of $L$ is $n$ (i.e $L^n\not = 0, L^{n+1}=0$). Is it correct that $K\cap L^n \not=0$?
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1answer
42 views

Whitehead's lemma (Lie algebras) for reductive Lie algebras.

I move the question here. Whitehead's lemma (Lie algebras) is: Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero, $V$ a finite-dimensional module over it and $f$: ...
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1answer
16 views

Selfnormalizing sub-algebra and direct sum decomposition

I got the following setting: Consider the decomposition $L=H+\sum_{\alpha\in \phi}L_{\alpha}$, where the sum is a direct sum, the $L_{\alpha}$ are the root spaces and $H$ is nilpotent (because its ...
2
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1answer
35 views

The center of nilpotent Lie algebra

Let $L$ be a non-abelian nilpotent Lie algebra and $Z(L)$ be its center. Is it possible for $Z(L)$ to be a maximal ideal in $L$? An attempt: Since $L$ is not abelian then there exists $x\in L, ...
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0answers
17 views

The Lie algebra quotient of a maximal ideal

If $\mathcal g$ is a Lie algebra, and $Z(\mathcal g)$ is its centralizer. Suppose that $Z(\mathcal g)$ is a maximal ideal of $\mathcal g$. Is it correct that $\mathcal g/Z(\mathcal g)$ is of dimension ...
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2answers
26 views

codimension of the derived algebra of a nilpotent Lie algebra

Let $\mathcal g$ be a nilpotent Lie algebra with $dim>1$. Why it is impossible that $codim \ \mathcal g'=1$?
2
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1answer
58 views

Is a real logarithm of a special orthogonal matrix necessarily skew-symmetric?

The exponential map from the Lie algebra of skew-symmetric matrices $\mathfrak{so}(n)$ to the Lie group $\operatorname{SO}(n)$ is surjective and so I know that given any special orthogonal matrix ...
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0answers
16 views

Realizing a reflection group as a Weyl group

Question 1. Suppose given a compact, connected Lie group $G$ and a subtorus $S$ (not maximal) such that the effective image $N$ of the $\mathrm{Ad}$-action of the normalizer $N_G(S)$ on the Lie ...
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1answer
39 views

Lie Bracket Calculation for Integral Curves

I am trying to derive a Lie bracket, and then find the related integral curve at the point $(x_0,y_0)$. The problem gives the vector fields $X = y \frac{\partial }{\partial x}$ ,$Y = \frac{x^2}{2} ...
0
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1answer
28 views

Isomorphic root systems?

I found two different root systems of $sl(3,\mathbb C)$ in the internet. The first one: And the second one: I think they should be the same. So...how can I see that these root systems are ...
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1answer
38 views

Understanding Quotienting by Relations vs Quotienting by Generators

I understand the idea of a quotient algebra $A / I$ where $A$ is a $K$-algebra and $I$ is a two-sided ideal, i.e. I understand the projection map as an algebra morphism. However, I'm unsure about how ...
2
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1answer
22 views

Cartan subalgebra of ${\frak so}(5)$

I would like to find effectively the Cartan subalgebra of ${\frak so}(5)$. Does anybody knows how to proceed? Edit: I don't want to start from the simple roots and then derive it. I would like to do ...
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1answer
30 views

Matrix representation of Lie Algebra $B_2$

I'm writing some practical examples where to calculate the Killing form, the Cartan Matrix, Dynkin diagrams etc. Does anybody have on or two nice matrix representations of the $B_2$ Algebra? It would ...
2
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1answer
67 views

The root system of $sl(3,\mathbb C)$

I want to determine the root-system of the lie algebra $sl(3,\mathbb C)$. Does someone know a good (and complete) reference for this problem? I know that the root-system is $A_2$ but I want to see a ...
0
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1answer
27 views

Relations between Lie algebras and Lie coalgebras.

Let $g^*$ be the dual vector space of a vector space $g$. Suppose that $g^*$ is a Lie algebra and $[,]_{g^*}: \Lambda^2 g^* \to g^*$ satisfies the Jacobi identity. Let $\delta: g \to \Lambda^2 g$ be ...
2
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1answer
40 views

Spheres as Symplectic Homogeneous Spaces

Does there exist a description of the odd dimensional spheres as homogeneous spaces of the symplectic group. For $S^7$ it seems to me that we should have $S^7 \simeq Sp(3)/Sp(2)$, but I can't make a ...
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0answers
33 views

Linear Algebraic Groups with Same Lie Algebra (Soft Question)

Let $G$ and $H$ be two linear algebraic groups over an algebraically closed field $F$ (char 0 ) such that their lie algebras are isomorphic. Now what can we say about the relation between these two ...
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0answers
35 views

Simply connected linear algebraic group [duplicate]

Following is what I understand regarding the simply connected linear algebraic groups afer reading some definition in Hochschild's 'Basic Theory of Algebraic Groups' : (I don't know about fundamental ...
2
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1answer
63 views

If the Lie algebra is ${\frak g}={\frak a}\oplus{\frak b}$ then the Lie group is $G=AB$?

Let $G$ be a connected Lie group and suppose that its Lie algebra ${\frak g}$ splits into a direct sum of ideals $${\frak g} = {\frak a}\oplus{\frak b}.$$ Let $A$ be the connected Lie subgroup of $G$ ...
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0answers
25 views

$so(4)$ is isomorphic to $so(3)+so(3)$.

Since every $4×4$ skew-symmetric matrix can be written uniquely as a decomposition $$\begin{bmatrix} 0&-a&-b&-c ...
0
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1answer
25 views

How does one compute the Hurewicz homomorphism for a (symplectic) nilmanifold?

I have a symplectic six-dimensional nilmanifold $X:=G/\Gamma$ in hand, characterized by the sextuple $(0,0,12,13,14+23,24+15)$, which records the exterior derivatives of a basis of $\Gamma$-invariant ...
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0answers
38 views

Lie algebra of a connected simple algebraic group is simple and a simply connected algebraic group having the same Lie algebra

Let $G$ be a connected simple algebraic group over an algebraically closed field $C$. What I infer from this definition is that the defining polynomials of $G$ have coefficients in $C$ while $G$ may ...
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2answers
54 views

Is the Lie bracket of two vector fields well defined?

I want to understand what exactly means to ask the question if the Lie bracket $[X,Y]$ of two vector fields $X,Y\in \mathcal{XM}$, where $\mathcal{M}$ is a differentiable manifold, is well defined. ...
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0answers
15 views

Derived Algebra is nilpotent implies the lie algebra is solvable [duplicate]

How does one show that Derived Algebra is nilpotent implies the lie algebra is solvable. My attempt: Let $L$ be such a Lie-algebra then $[L,L]$ is nilpotent so it is solvable. So $[L,L]^{(n)}=0$ for ...
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2answers
33 views

Ideals of $\mathfrak{gl}_n$

How does one determine the ideals of $\mathfrak{gl}_n(C)$? My guess is that the only ones are $(0) $ and $\mathfrak{sl}_n(C)$. I think approaching the problem by the fact that each $\mathfrak{g}^{ ...
0
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2answers
32 views

On the meaning of the word “generic” in Lie Algebra (or otherwise)

I always have a problem with the word generic in the literature of mathematics. Let me ask you a specific question about "non-degenerate $\mathbb{Z}$-graded lie algebras''. The definition I'm working ...
2
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1answer
47 views

Structure theorem for non connected graded Hopf algebras

Let $V^{\bullet}$ be a graded vector space. Assume that $V^{i}=0$ for $i<1$. Then the tensor algebra has the structure of a connected Hopf algebra and in particular its Lie algebra of primitive ...
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1answer
40 views

Center of $\mathfrak{gl}(n,\Bbb F)$ using the adjoint representation

I'm quite new to Lie Algebras, and so there's a lot of easy stuff that I'm probably missing. Anyway following Kac notes I'm asked to compute the center of $\mathfrak{gl}(n,\Bbb K)$, and I've done it ...
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0answers
31 views

Are there proper Ad invariant sets on simple lie algebras?

Let $\mathcal C\subset \mathfrak g$ be a subset in a Lie algebra $\mathfrak g$ satisfying the following two conditions: $\mathrm{Ad}(G)\mathcal C=\mathcal C$ If $X,Y\in \mathcal C$, then ...
4
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1answer
56 views

For which Lie groups $G$ can one write $g$ as the exponential $\exp X$ of some $X \in {\frak g}$ for every element $g \in G$?

I am reading a book on matrix Lie algebras (Brian Hall's). Corollary 2.30. says that if $G$ is a connected matrix Lie group, then every element $A$ of $G$ can be written in the form ...
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0answers
37 views

Group action of linear algebraic group $G$ on itself induces a representaion of $G$ on $Lie(G)$

Let us be given a linear algebraic group $G$ over a field $K$ of characterstic zero. This group $G$ is defined as the common zeroes of a finite set of polynomials $\{f_1, \ldots ,f_r\}$ $\in K ...
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1answer
37 views

Action of an algebraic group induce a representation of its Lie algebra

Let $G$ be a linear algebraic group over a field $K$ of characterstic zero acting on a vector space $V$. Then does this action induce a representation : $$\Gamma : Lie(G) \to gl(V)$$ If yes, how ? ...
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0answers
47 views

Some doubts on the relationship between Lie algebras and Lie groups

Let $(\mathbb G,*)$ be a Carnot group. Thus, by definition, $\mathbb G$ is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ admits a stratification, that is $$\mathfrak ...
2
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1answer
65 views

Coset Space as a Representation of a Lie Algebra

I'm reading through some notes (about the use of Lie groups/algebras in physics) obtained from a friend from a class that took a while back, and I can't quite figure out where one thing regarding some ...
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1answer
12 views

On the assumptions of cocyle conditions in a Lie algebra

To define the Cohomology (with values in $\mathbb{C}$) on a lie algebra $L$, we define a coboundary map $\delta:\Lambda^n(L)\to \Lambda^{n+1}(L)$. There is a general formula for the coboundary map but ...
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0answers
12 views

Understanding if the definition of constant normal set depends on the choice of the scalar product or not

Suppose we have a Lie group on $\mathbb R^n$, let's say $(\mathbb R^n,*)$. Suppose also that its Lie algebra $\mathfrak g$ is stratified: I mean that there exists a decomposition of $\mathfrak g$ as ...
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1answer
21 views

If x is in the derived algebra, show that Tr(ad a)=0 [closed]

Let $L$ be a Lie algebra, and let $a\in [L,L]$. How to prove that $trace(ad_a)=0$?
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55 views

Open problems in Lie theory

I been studying lie theory for some time. Beside classification related problems what are some examples of open problems in the lie world? Especifically in the topological/differentiable structure of ...
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15 views

Product of root multiplicities in Kac-Moody algebras

Let $\mathfrak{g}$ be a Kac-Moody Algebra with GCM $A$. Let $\alpha$ and $\beta$ be two roots not necessarily real and $g_\alpha$ and $g_\beta$ be the corresponding weight spaces of dimension ...
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0answers
46 views

Finding the radical of $\mathfrak{gl}(2,\mathbb{C})$ [duplicate]

I am taking a Lie algebras course as a prerequisite to study Lie groups. The idea of a radical of a Lie algebra (maximal solvable ideal) has been defined in class but no other statements or theorems ...
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0answers
23 views

highest weight of adjoint represesentation

Let $\mathfrak{g} = \mathfrak{gl}(3,\mathbb{C})$ and let $\mathfrak{h}$ be the subalgebra of $\mathfrak{g}$ consisting of diagonal matricies. Then for $1 \leq i \leq n$, let $\epsilon_i \in ...
3
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1answer
34 views

Another way of describing a maximal torus

Consider the Lie group $SU(2)$. A maximal torus for $SU(2)$ is $$T=\left\{\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{-i\theta}\end{pmatrix}:\theta\in{\Bbb R}\right\},$$ and its Lie algebra is ...
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1answer
34 views

Connection between quivers and representations of Lie algebras

Can anyone recommend a reference to study the connection between quiver theory and representation theory of Lie algebras? Supposedly those two things have something to do with each other, with the ...
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2answers
52 views

Computation of killing form

The killing form is denoted by $ B $. We know that for all $ X,Y \in gl\left(n,\mathbb{R}\right) $ $$ B\left(X,Y\right)=2n\ tr\left(XY\right)-2\ tr\left(X\right)tr\left(Y\right) $$ So for $ X,Y \in ...