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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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
34 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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Under certain conditions the exponential map is surjective

Let $\textbf{x}\in SL \left(2,\mathbb{R}\right) $. Show that: $\textbf{x}=\exp\left(X\right)$ for some $X \in sl\left(2,\mathbb{R}\right)$ if and only if $\textbf{x}$ have positive real proper values ...
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12 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
25 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
28 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
27 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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27 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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1answer
47 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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31 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
35 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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35 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
18 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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43 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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14 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
28 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
53 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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39 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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18 views

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
22 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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22 views

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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1answer
34 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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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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0answers
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
37 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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2answers
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 ...
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2answers
36 views

Simple 3-dim Lie algebra

If $g$ is a Lie algebra, how to prove that $Tr(ad \ a)=0$ for all $a\in [g,g]$? In case $dim\ g=3$ and $[g,g]=g$ how to show that $g$ is simple?
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24 views

Why is the sign of $L^2$ opposite?

The following figure is from an article about Casimir element of Wikipedia. According to the following, $L_x^2$ is not a simple matrix multiplication of $L_x$ with itself, since the sign is opposite. ...
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More Hopf algebra confusion: Verifying an equation between matrix coefficients.

I am thinking of the following situation: The lie algebra $g = sl_2(\mathbb(C))$, and $V(1)$ is the unique dimension 2 irreducible representation (the defining representation). Let $U$ be the ...
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2answers
38 views

Show that every derivation of the non-abelian two-dimensional Lie algebra is inner

For any $2$-dimensional non-abelian Lie algebra $g$, there exists a basis $a,b$ such that $[a,b]=a$. Now I want to prove that any derivation of $L$ is inner. My proof: $D[x,y]=[Dx,y]+[x,Dy]= [k_1 ...
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1answer
32 views

$V(1)$ generates the tensor category of representations of $sl_2(\mathbb{C})$ - what exactly does this mean?

Does anyone know what it means for a $V(1)$ to generate the tensor category of (finite dimensional) representations of $sl_2(\mathbb{C})$. Here $V(1)$ is the 2 dimensional irreducible Lie algebra ...
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1answer
62 views

Regarding the representation theory of $SL_2(\mathbf{R})$.

Dear friends of mathematics, I have the following question for you. (a) According to Wikipedia there is a unique irreducible (real??) $2$-dimensional representation of $SL_2(\mathbf{R})$, which must ...
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18 views

How to compute the center of the universal enveloping algebra of a simple Lie algebra

Given a simple Lie algebra $\mathfrak g$ over $\mathbb C$. Then the center $Z(\mathfrak g)$ of the universal enveloping algebra of $\mathfrak g$ is a polynomial algebra in $l$ generators, where $l = ...
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53 views

A general definition for a character of a (not necessarily associative) algebra

Let $A$ be a algebra over a algebraically closed field $k$. Is there certain definition of a "character" $f: A \rightarrow k$? That is, what is the common and useful condition for a linear map $f: A ...
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25 views

How to compute Casimir elements of $g \otimes g$?

Let $g$ be a Lie algebra. How to compute Casimir elements of $g \otimes g$? I am asking this question because in the book a guide to quantum groups, page 80, there is an equation $r_{12} + r_{21}=t$, ...
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1answer
13 views

Universal enveloping algebra, simple algebraic identity $[E,EF] = [E,E]F + E[E,F]$?? (in $sl_2(C)$)

Reading some notes on Lie algebras now. Let $g$ be the Lie algebra $sl_2(\mathbb{C}))$ and let $U$ be the universal enveloping algebra. In the course of some computation (proving the Casimir element ...