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 homomorphism preserves Jordan form

Fact : $\phi : L_1\rightarrow L_2$ is $surjective$ Lie algebra homomorphism. If $h\in L_1$ and ${\rm ad}_h$ is diagonalizable then ${\rm ad}_{\phi(h)}$ is diagonalizable ...
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56 views

Proof of Jordan Theorem on Lie Algebra

$ L$ is semisimple Lie algebra in ${\rm gl}\ V$ where $V$ is a complex vector space. Then we have two Jordan forms $$ x=d+n,\ {\rm ad}_x = {\rm ad}_d + {\rm ad}_n\ (x\in L) $$ (cf. Theorem 9.15 in ...
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176 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
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151 views

How do I find the Cartan subalgebra?

I know the definition of a Cartan subalgebra, but how do I actually find it explicitly for a particular Lie algebra over the complex numbers?
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92 views

Status of a question from Freeman Dyson's 1972 article

In a famous article, Freeman Dyson mentions an interesting relationship between the $\tau$ functions of number theory and the dimensions of finite-dimensional simple Lie algebras (section 2). He ...
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137 views

Vector fields on smooth manifolds and Lie algebras

I'm currently studying differential geometry on smooth manifolds using differential forms and I'm trying to apply it to what I have learned earlier about Lie groups, but something doesn't seem to ...
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111 views

Open questions in theory of Lie groups and Lie algebras

I am just about to finish an introductory book 'Lie groups, Lie algberas & Representations' by Brian Hall and am curious to know what are the current directions of research in this area. I learn ...
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81 views

How can I show that $\mathfrak{sl}_n(\mathbb{C})$ is a simple Lie algebra?

The question is in the title: how can I show $\mathfrak{sl}_n(\mathbb{C})$ is simple? In every book I scoured, they say $\mathfrak{sl}_n(\mathbb{C})$ is simple but they do not provide a proof! Is ...
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70 views

Complete reducibility of tensor product

Let $L$ be a Lie algebra (over a algebraically closed field, not sure if it is relevant). If $V$ and $W$ are two completely reducible $L$-modules, can anyone give a hint on how to show that $V\otimes ...
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71 views

Dynkin diagram construction

My question is how to construct the Dynkin diagrams of a semi-simple Lie group $G$, which is the product of simple Lie groups. Is it the combinaison of Dynkin diagrams of these simple Lie groups? For ...
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40 views

Example showing that the product of ideals must be the span of the commutators

I'm trying to find an example showing why, in a Lie algebra, we can't just define the product of two ideals $I$ and $J$ to be the elements of the form $[x,y]$ where $x \in I, \; y \in J$. I imagine ...
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129 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let G be an affine group scheme, and Dist(G) its hyperalgebra. I am wondering what is the relationship between Dist(G) and G interms of Cohomology? Is there a cohomology theory for Dist(G), if so ...
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52 views

Relationship between invariants of a simple algebraic group

Let $G$ be a simple algebraic group over an algebraically closed field $k$. I believe all of the following invariants are well-defined. Besides the coxeter number, I haven't read about the others, ...
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40 views

explicit realization of irreducible representations of simple lie algebras

I know explicit realization of irreducible representations of simple lie algebra $sl_n$ when the highest weight of that representation is a fundamental weight.Is there any explicit realization of any ...
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133 views

Root-subspaces are $\mathfrak{g}$-invariant

I need some help with technicalities concernig root-subspaces of nilpotent Lie algebras of operators. Let $\mathfrak{g} \subseteq \mathfrak{gl}(V)$ be nilpotent Lie algebra, $\alpha \ \colon \ ...
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36 views

Classification of 6-D Nilmanifolds

I am reading the G.Cavalcanti and M.Gualtieri's Generalized Complex Structures on Nilmanifolds. In the introduction it is said that there are 34 nilpotent lie algebra isomorphism classes. There are ...
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72 views

Cartan's Criterion. $L$ solvable $\implies$ $Tr(xy) = 0$ $\forall x \in L$, $\forall x \in L^{(1)}$

Cartan's Criterion. Given $V$ a finite dimensional complex vector space and $L$ a Lie subalgebra of $gl(V)$ then, $L$ solvable $\implies$ $Tr(xy) = 0$ $\forall x \in L$, $\forall x \in L^{(1)}$. ...
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52 views

Prove that $[L,Rad(L)] \subseteq N$ for finite-dimensional Lie algebra $L$

I need to prove the following fact: if $L$ is a finite-dimensional Lie algebra over field of characteristic $0$, $Rad(L)$ is its radical, and $N$ is the maximal nilpotent ideal in $L$, then ...
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73 views

Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)

This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups. More precisely, given discrete groups below (a), (b), (c): ...
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222 views

Semi-simple Lie algebra $L$ coincides with its derived algebra $L'$

If $L$ is a semi-simple Lie algebra, then $L=L'$. Since $L$ is semi-simple we can write it as a direct sum of simple ideals $L_i$, i.e. $L=\oplus_{i=1}^r L_i$. Then $L'=\oplus_{i=1}^r L_i'$ and ...
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42 views

To what extent are formulas obtained in one Lie group valid in another Lie group with an isomorphic Lie algebra?

In quantum optics, I am trying to explore the group generated by squeezing and rotation operators. These are closely related to area-preserving linear transforms, which they induce on the phase space, ...
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105 views

Character of half-spin representation

Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum $$\sum x_1^{\pm ...
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202 views

Determinant of a cartan matrix

I was taking an introductory course in Lie algebras and I just learned about how we associate a Cartan matrix to a semisimple Lie algebra. So, for the A-series, the determinant of this matrix goes to ...
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24 views

Finding the Lie map

Suppose I have a group homomorphism $\rho:SL(2,\mathbb{C})\to SO_0(3,1)$ given by $\rho(a)X=aXa^*$ and I want to see how the corresponding Lie map $L\rho$ looks like. By definition $$ ...
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33 views

Only 3 finite-dimensional Lie algebra on $\mathbf R$?

Please, how does one show that up to diffeomorphism there are exactly three finite dimensional Lie algebras of vector fields on the real line $\mathbf R$.
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100 views

If $\mathfrak{g}$ admits a decomposition then it is semisimple

I want to show: If $\mathfrak{g}$ is a Lie algebra that has an abelian subalgebra $\mathfrak{h}$ such that $\mathfrak{g}$ has a Cartan decomposition $\mathfrak{g}=\mathfrak{h}\oplus(\bigoplus_{\alpha ...
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33 views

Integer domain enveloping algebra

I must prove that if $L$ is a Lie algebra and denoting $U(L)$ the enveloping algebra, then $U(L)$ hasn't zero divisions (e.g. if $ab=0 \,\,\, a,b \in U(L)$ then $a=0$ or $b=0$). Some ideas?
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43 views

Symmetric algebra and complex polynomial on Lie algebra

Let $G$ a Lie group and $\mathfrak{G}$ its Lie algebra. How can I identify the symmetric algebra on $\mathfrak{G}$ ($S(\mathfrak{G})$) with the algebra of complex polynomials on $\mathfrak{G}$, that I ...
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89 views

$x$ regular $\Leftrightarrow$ $x$ is in exactly one CSA

Here's a statement and a proof given in a Lie Algebra course (in the tutorial): Let $L$ be a semisimple Lie algebra over a field $F$ with $\text{char} F=0$. Let $x\in L$ be a semisimple element. ...
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62 views

Finite-dimensional Lie algebra as a scheme

Kindly asking for any hints about the following questions: Suppose $k$ is an algebraically closed field of characteristic zero and $g$ is a finite-dimensional Lie algebra over $k$. Then $g$ is ...
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81 views

Dimension of Abelian Lie Algebras

I've tried to answer this question, but I need some help. What is the possible dimension of irreducible representations of Abelian Lie Algebras? I think it is always one, but I am not sure. Thank ...
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91 views

Commutator formula in infinite dimensions

The commutator formula states that for $A,B$ elements of a Lie algebra, $$ \lim_{n\to \infty}\left\{ ...
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41 views

Smallest dimensional irreps of semi-simple Lie algebras

I'm wondering if there is a reference that lists the first couple smallest dimensional irreducible representations of each semi-simple Lie algebra. I know these can be found using the Weyl dimension ...
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301 views

Standard parabolic Lie subalgebras and conjugacy

Let $\mathfrak g$ be a given semisimple Lie algebra with corresponding adjoint Lie group $G$. A parabolic subalgebra is any subalgebra containing a Borel subalgebra. We can pick a Borel ...
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65 views

ADE type root lattice

Let $\Phi$ be a root system of ADE type, $L$ is the corresponding root lattice, show that $\Phi=\{\alpha\in L:(\alpha,\alpha)=2\}$, where $(,)$ is the normalized non-degenerate symmetric bilinear form ...
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294 views

Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
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69 views

Basis of the Engel algebra

If I have a connected, simply connected nilpotent lie group given by the commutators between the elements of a basis of its Lie algebra how can I recover the left invariant vector fields? For ...
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144 views

Finding the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$

How can I find the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$? I'm explicitly working with basis vectors in trying to compute $\operatorname{tr}(\operatorname{ad}(a)\operatorname{ad}(b))$ but ...
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69 views

Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.

Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in ...
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120 views

Question about a Corollary of Engel's Theorem

Engel's Theorem states that: Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V \neq 0$, then there exists nonzero $v \in V$ for ...
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80 views

Principal Bundles and Lie algebras

Suppose I have a principal $S^{1}$ bundle over a nice compact symmetric space. The symmetric space arises as a homogeneous space, call it $X=G/H$. On the Lie algebra level we have the decomposition ...
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45 views

Solve commutator relation $[Q,d]=-[P,d]$ for $Q$ on chain complexes with scalar product

Suppose we are given chain sequences $\dots \rightarrow C_k \rightarrow C_{k+1} \rightarrow \dots$ and $\dots \rightarrow D_k \rightarrow D_{k+1} \rightarrow \dots$ of finite-dimensional vector ...
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76 views

orbit of a Dynkin diagram automorphism

Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$. What is a set of representatives of the orbits of $\Delta$ under $f$ ? Thanks,
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25 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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27 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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42 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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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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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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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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31 views

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 ...