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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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 ...
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27 views

Are there some new “function or even topic” in lie theory with special functions? [on hold]

Every one: I research in Lie theory with special functions. But I saw a lot of research for most of the special functions and polynomials. I wish you could recommend a specific kind of these special ...
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1answer
42 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 ...
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12 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 ...
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1answer
25 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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23 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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26 views

Simply connected linear algebraic group

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 ...
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11 views

Finding Kac-Moody algebras with given root strings

For two hypothetical real roots $\alpha$,$\beta$ and their hypothetical root strings $S(\alpha, \beta)$ and $S(\beta,\alpha)$ through each other, is there a general procedure (or method of "informed ...
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1answer
50 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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21 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 ...
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1answer
17 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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32 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
43 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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14 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
26 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}^{ ...
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2answers
30 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 ...
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1answer
28 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
36 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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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 ...
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48 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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32 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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44 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 ...
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46 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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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
19 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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45 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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41 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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22 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 ...
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1answer
30 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
28 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
46 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 ...
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1answer
67 views

Every automorphism of $\mathfrak{sl}(2,\mathbb{C})$ is given by conjugation with some $u \in \mathrm{SL}(2,\mathbb{C})$

Let $\mathfrak g = \mathfrak{sl}(2, \mathbb{C})$. Let $\gamma \in \operatorname{Aut}(\mathfrak{g})$. How to show that $\gamma$ is conjugation by some $u \in \mathrm{SL}(2, \mathbb C)$?
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40 views

Matrix representation of a 6-dimensional Lie algebra

The question is about the matrix representation of the following 6-dimensional Lie algebra, with 6 generators $t_1,t_2,t_3,t_4,t_5,t_6$. This Lie algebra is nilpotent, non-abelian, non-reductive and ...
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1answer
86 views

Computing the radical of $\mathfrak{gl}(2,\mathbb{C})$ without using the semisimplicity of $\mathfrak{sl}(2,\mathbb{C})$.

I have been trying to show that the radical of $\mathfrak{gl}(2,\mathbb{C})$ is its center, i.e. scalar matrices, however all the proofs I have encountered (e.g. Radical of $\mathfrak{gl}_n$) have ...
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How is vector, dual vector, etc defined in Matrix Lie Group Manifold?

How is vector, dual vector, etc defined in Matrix Lie Group Manifold? Are the coefficients matrices and the (dual)basis matrices as well?For example, the Maurer–Cartan form can be written as ...
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19 views

Lie element in a non-commutative algebra?

While I am reading a paper, there is a weird, at least for myself, notion I have never seen: let $R:=\mathbb{Q}_\ell\{\{X,Y\}\}$ be a $\mathbb{Q}_\ell$-algebra of formal power series in non-commuting ...
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35 views

determining whether Lie algebra is enlarged by new generator

Is there a simple criterion to determine whether, given a set of generators $A_1,\ldots,A_n$ of a Lie subalgebra $\mathfrak{h}\subset\mathfrak{g}$, adding a new element $B\in\mathfrak{g}$ will enlarge ...
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1answer
23 views

Left multiplication isometry?

If $G$ is a semi-simple Lie group and $g\in G$, then $G$ has a bi-invariant metric which is a Riemmanian metric. My question is: with respect to this metric does the left-translation map ...
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1answer
51 views

Exercise on representations of $\mathfrak{sl}_2 \mathbb{C}$ (Etingof 1.55.j)

I'm working through these notes on representation theory: http://math.mit.edu/~etingof/replect.pdf. Currently I'm looking at exercise 1.55 part j. I've done all of the previous parts and most of the ...
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Lie algebras isomorphism $gl(2,\mathbb{C}) \cong sl(2,\mathbb{C}) \oplus \mathbb{C}$ [duplicate]

Having a struggle finding a lie algebra homomorphism between these two spaces. So i know that $\mathbb{C} \cong \lambda\left( \begin{array}{ccc} 1 & 0 \\ 0 & 1 \\ \end{array} \right) $, ...
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1answer
34 views

Compactification of Lie Group

Is there a way to embed a Lie Group $G$ into a compact lie Group $H$, such that the inclusion is a Lie group homomorphism?
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38 views

Particular on the structure of a weight $L$-module $M$, with $L$ semisimple Lie-algebra.

Let be $L$ a semisimple Lie-algebra with its root system $R \subset H^*$ Cartan subalgebra and root decomposition \begin{gather} L=H \oplus \bigoplus_{\alpha \in R}^n L_{\alpha} \end{gather} Let $M$ ...
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2answers
50 views

Defining Unitary Matrices

I have got a question and I would appreciate if one could help. I start with an example to explain what I am looking for. Assume a scaled unitary matrix like $U_2 = \begin{bmatrix} 1 & 1 \\ 1 ...
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30 views

3-dim Lie algebras

If $\mathfrak g$ is a three dimensional Lie algebra and $[\mathfrak g,\mathfrak g]=\mathfrak g$. How to prove that there is a basis $\{x,y,z\}$ such that either $[x,y]=z, [y,z]=x, [z,x]=y$ or ...
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1answer
38 views

Higher self-extension $\text{Ext}^i_{\mathcal{O}}(L(\lambda), L(\lambda))$ between two irreducible modules in BGG category $\mathcal{O}$

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $\mathcal{O}$ be the BGG category for $\mathfrak{g}$. It is well-known that the set of irreducible ...
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2answers
41 views

3-dim simple complex Lie algebra

If we know that a 3-dimensional Lie algebra $L$ with $[L,L]=L$ is simple. How to prove that the only (up to isomorphism) 3-dimensional complex Lie algebra $L$ with $L=[L,L]$ is $sl_2(\mathbb C)$?
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37 views

Riemannian metrics on homogeneous spaces

Let G be a Lie group and H be a compact subgroup. The (left) coset space G/H is, up to an isomorphism, equivalent to the smooth homogeneous manifold M. My question is, is it possible to impose an ...