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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5
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1answer
152 views

Subspace spanned by eigenvectors is a subalgebra

Let $L$ be a Lie algebra over an algebraically closed field and let $x\in L$. Prove that the subspace of $L$ spanned by the eigenvectors of $\operatorname{ad}x$ is a subalgebra. Suppose the ...
2
votes
0answers
56 views

Isomorphism between $B_2$ and $C_2$ [duplicate]

For small values of $l$, isomorphisms occur among certain of the classical algebras. Show that $B_2$ is isomorphic to $C_2$. Well, both $B_2$ and $C_2$ have dimension $10$. $B_2$ consists of ...
0
votes
1answer
38 views

$L=[LL]$ for classical algebras

When $\operatorname{char}F=0$, show that each classical algebra $L=A_l,B_l,C_l$ or $D_l$ is equal to $[LL]$. Since $L$ is a Lie algebra, we know that if $x,y\in L$, then $[x,y]=xy-yx\in L$. So ...
4
votes
1answer
110 views

Eigenvalues of $\operatorname{ad}x$

Let $x\in \operatorname{gl}(n,F)$ have $n$ distinct eigenvalues $a_1,\ldots,a_n$ in $F$. Prove that the eigenvalues of $\text{ad }x$ are precisely the $n^2$ scalars $a_i-a_j$ ($1\leq i,j\leq n$), ...
0
votes
3answers
114 views

Linear Lie algebra isomorphic to two dimensional algebra

Find a linear Lie algebra isomorphic to the nonabelian two dimensional algebra with basis $x,y$ such that $[x,y]=x$. (Hint: Look at the adjoint representation.) $\DeclareMathOperator{\ad}{ad}$The ...
1
vote
2answers
72 views

Lie algebra of dimension 2

From Humphreys' Introduction to Lie Algebras and Representation Theory: We can determine (up to isomorphism) all Lie algebra of dimension $2$. Start with a basis $x,y$ of $L$. Clearly, all ...
2
votes
1answer
67 views

One-dimensional Lie algebra with non-trivial bracket operation

We can define a Lie algebra letting $\mathbb{R}$ be the vector space and also the field. We can then have $[x,y]=xy-yx=0$ for all $x,y$. Is there a one-dimensional Lie algebra such that $[x,y]$ is ...
8
votes
5answers
407 views

Getting started with Lie Groups

I am looking for some material (e.g. references, books, notes) to get started with Lie Groups and Lie Algebra. My motivation is that I (eventually) want to understand the theory underpinning papers ...
2
votes
1answer
217 views

Is the Lie bracket of a Lie Algebra of a Matrix Lie Group always the commutator?

Given a Matrix Lie Group, the Lie Bracket is of the associated Lie Algebra is given by the Lie Derivative. Is this always the commutator if we start from a Matrix Lie Group? Cheers!
1
vote
1answer
54 views

$A$ as Lie algebra Jacobi identity

In page 2 of Hans Samelson's Notes on Lie Algebras, the text gives an example of Lie algebra $A_L$. Let $A$ be an algebra over $\mathbb{F}$ (a vector space with an associative multiplication ...
6
votes
2answers
281 views

Learning representation theory of Lie groups for someone who knows Lie algebras

I'd like to learn the representation theory of Lie groups. I have a good knowledge of semisimple Lie algebras and their representation theory as well as the basics of Lie groups. To what extent are ...
3
votes
1answer
150 views

Computing the Killing form.

Let $\mathfrak{g}$ be a finite dimensional Lie algebra. The Killing form $K:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{C}$ is given by $$K(x,y) = tr(ad_xad_y)$$ I have two questions about the ...
1
vote
1answer
331 views

Lie algebra of normal subgroup is an ideal

I want to prove that if $G$ is a connected Lie group, $H$ is a normal Lie subgroup of $G$, $\mathfrak{g}$ and $\mathfrak{h}$ their respective lie algebras, then $\mathfrak{h}$ is an ideal of ...
1
vote
0answers
77 views

Lorentz group and eigenvalues

For generators of the Lorentz group ($\hat {R}_{k}$ corresponds to the generators of 3-rotations, $\hat {L}_{k}$ corresponds to the generators of the boosts) we have the following algebra: $$ [\hat ...
2
votes
0answers
74 views

Does the Lie Bracket automatically exist?

Let $g$ be a Matrix Lie Group. The Lie Algebra of $g := Lie(g)$ is defined as $ Lie(g) = \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1, \gamma(0) = \mathbb{I} \} $ ...
2
votes
0answers
27 views

When is $\left\{ [x,y] \mid x, y \in \mathfrak{g} \right\}$ a subspace of $\mathfrak{g}$?

Let $\mathfrak{g}$ be a lie algebra. When is $\left\{ \left[x,y\right] \mid x, y \in \mathfrak{g} \right\}$ a subspace of $\mathfrak{g}$? Is this common at all? Thanks! -Dan
0
votes
0answers
67 views

Calculating an expression for the trace of generators of two Lie algebra.

Suppose we have $$[Q^a,Q^b]=if^c_{ab}Q^c$$ where Q's are generators of a Lie algebra associated a SU(N) group. So Q's are traceless. Also we have $$[P^a,P^b]=0$$ where P's are generators of a Lie ...
1
vote
1answer
56 views

The tangent space of $\mathrm{Aut}(T_eG)$

Let $G$ be a Lie group and $e \in G$ be the identity. I want to understand the following sentence. " $\mathrm{Aut}(T_eG)$ being just an open subset of the vector space of endomorphisms of $T_eG$, its ...
3
votes
2answers
55 views

Metric over a Lie algebra $\mathfrak{u}(n)$

Let $\mathfrak{u}(n)$ be the Lie algebra of the Lie group $U(n)$. I can define a positive-definite inner product over $\mathfrak{u}(n)$ in this way: if $A,B \in \mathfrak{u}(n)$ I define $\langle A,B ...
1
vote
1answer
80 views

representations and modules

I am reading representations of Lie Algebra in Humphreys.He is defining representation as $L$-modules. In case of group representation we have the correspondence between representation and modules ...
8
votes
3answers
227 views

Representing $\mathbb{R}/\mathbb{Z}$ as a matrix group.

It was told to me that $G = \mathbb{R}/\mathbb{Z}$ is a real matrix group. Can someone help me understand how to represent $G$ in $Gl_n(\mathbb{R})$ for some $n$? (Supposedly, $n = 1$? But that's ...
2
votes
1answer
117 views

Finding the lie algebra of the symplectic lie group

I am having difficulties completing my proof that $\text{Lie}(\text{Sp}(2n)) \equiv \mathfrak{sp}(2n) = \{ X \in Gl(2n)\; |\; X^TJ + JX = 0 \}$ Where $J \equiv \begin{bmatrix}0 & \mathbb{1}_n ...
0
votes
1answer
40 views

Does $\mathrm{Im}(\exp)$ being a manifold imply the domain is a manifold?

Let $G$ be a real matrix group of dimension $n$. Let $\mathfrak{g}$ denote the lie algebra of $G$. Suppose $X \subset \mathfrak{g}$ such that $e^X$ is a submanifold of $G$ of dimension $m$. Does ...
5
votes
0answers
42 views

Does every element of $G_2 = \mathrm{Aut}(\mathbb{O})$ stabilize a quaternion subalgebra?

Has anyone heard of this result before? Let $\mathbb{O}$ denote the octonions and let $G_2$ denote its automorphism group (i.e. the 14 dimensional subgroup of $SO(7)$). Then any element of $G_2$ ...
7
votes
1answer
268 views

What are spinors mathematically?

In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing. There are essentially two frameworks for viewing the notion of a ...
0
votes
0answers
92 views

Which Lie group / algebra is generated by these three matrices?

This is a beginner question (and not any homework). I want to get a feeling for Lie group/algebra generators. Do the three matrices $$A=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0& ...
5
votes
1answer
628 views

What are defining & fundamental representations?

In physics terminology, one hears of the fundamental & defining representations of lie algebras or groups - are these the same as irreducible representations?
5
votes
1answer
451 views

Expression for the Maurer-Cartan form of a matrix group

I understand the definition of the Maurer-Cartan form on a general Lie group $G$, defined as $\theta_g = (L_{g^{-1}})_*:T_gG \rightarrow T_eG=\mathfrak{g}$. What I don't understand is the expression ...
3
votes
2answers
93 views

Equivalent definitions of Verma modules

This is a rather basic question. I was reading some notes on geometric representation theory by Gaitsgory and his defition of Verma module is the following: Let $ \lambda $ be a weight of $ ...
2
votes
0answers
123 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 \ ...
0
votes
1answer
92 views

anomalous minus sign in commutator of vector fields

Define $$X_A(x):=A^i_{\ j}x^j$$ where $A$ is a matrix. Why is there a minus sign in the following formula? $$[X_A,X_B]=-X_{[A,B]}$$ Edit: perhaps the question is not well posed, since what I really ...
1
vote
2answers
84 views

behavior of scalar product defined by trace under commutator

Define $$\langle X,Y \rangle := \operatorname{tr}XY^t,$$ where $X,Y$ are square matrices with real entries and $t$ denotes transpose. I have some troubles in proving that $$ \langle [X,Y],Z \rangle = ...
3
votes
0answers
71 views

A Isomorphism between the extension group and cohomology group of Lie algebras

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove ...
3
votes
0answers
55 views

PBW theorem for restricted Lie algebras

I'm looking at the proof of the PBW theorem for restricted Lie algebras to be found in Ponto and May's "More Concise Algebraic Topology", page 361 (367 in linked file). I either see an error in their ...
0
votes
2answers
80 views

How to work out the inverse matrix $A^{-1}$ ?

Suppose A is a matrix over some ring R (might be non-commutative). How to work out the inverse matrix $A^{-1}$?
1
vote
0answers
57 views

writting a code for finding the Kostant partition function

How to write a code in sage for finding the Kostant partition function for the elements of root lattice of rank 1 affine lie algebra $A_{1}^{(1)}$ which is defined as follows: $K(\beta)$ = the ...
3
votes
1answer
120 views

Weyl group of a restricted root system

What is the order of the Weyl group of the restricted root system of the real Lie algebra $\mathfrak g= \mathfrak{so}(p,q)$? More precisely, $\mathfrak g= \mathfrak k \oplus \mathfrak p$ and let ...
0
votes
1answer
52 views

Determine a basis for the Lie-Algebra $\text{sp}(\text{2n},\mathbb{C})$

Consider the Lie Group $\text{Sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}\mid\ J=g^TJg\}$ where $J=\begin{pmatrix} 0 & 1_n \\ -1_n & 0 \end{pmatrix} $. The corresponding Lie Algebra is ...
4
votes
0answers
87 views

Cartan Subalgebra and regular elements

Let $L$ be semisimple Lie algebra, $x\in L$ semisimple. Prove that if $x$ lies in exactly one Cartan subalgebra, then $x$ is regular.
3
votes
1answer
55 views

Proof that $\mathbf{R}[\omega]_\times\mathbf{R} = [\mathbf{R}\omega]_\times$

I have to prove that $$\mathbf{R}[\omega]_\times\mathbf{R}^\mathrm{T} = [\mathbf{R}\omega]_\times$$ Herein $\omega$ is a vector with elements. The notation $[\mathbf{a}]_\times$ is a conversion of ...
4
votes
2answers
438 views

Relation between root systems and representations of complex semisimple Lie algebras

I'm trying to understand the machinery of root systems for the purpose of classifying complex semisimple Lie algebras. During this process i lost the overview, espacially when it came to highest ...
1
vote
1answer
78 views

Weight spaces of Verma modules

Let $\mathfrak g$ be a semisimple Lie algebra generated by $x_i^+,x_i^-$, $1\leq i\leq n$, via the Chevalley-Serre relations and let $V(\mu)$ be a Verma module with highest-weight $\mu$. I gather that ...
0
votes
1answer
42 views

relation between the Poincaré and Euclidean algebra

Take $d$ a strictly positive integer, and consider the (proper) Euclidean group $E^d$ (the symmetry group of $\mathbf{R}^d$ with the conventional inner product), and the (proper, ortochronous) ...
4
votes
2answers
269 views

Lie algebra isomorphism between ${\rm sl}(2,{\bf C})$ and ${\bf C}^3$

I think that this is an exercise. I can not find a solution. We can define Lie bracket multiplication on ${\bf C}^3$ : $$ x\wedge y $$ where $x=(x_1,x_2, x_3)$, $y= (y_1,y_2,y_3)$, and $\wedge $ is ...
2
votes
2answers
113 views

Enveloping Algebra $U(L \oplus L')$

I'm having trouble understanding part of a proof of the following statement Let $L,L'$ be Lie algebras and $L \oplus L'$ their direct sum. Then $$ U(L \oplus L') \cong U(L) \otimes U(L')$$ Let ...
1
vote
1answer
94 views

$\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group?

I try to prove $\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group. My idea is that since $\exp\colon \mathfrak{sl}(2,\mathbb{R}) \to ...
4
votes
3answers
114 views

What does boson-type realization mean?

I have seen several different contexts the expression "boson-type realization", for instance in the study of algebras growth and realization of affine algebras. To be or not be a boson-type ...
4
votes
1answer
82 views

Infinitesimal $SO(N)$ transformations

An infinitesimal $SO(N)$ transformation matrix can be written : $$R_{ij} = \delta_{ij}+\theta_{ij}+O(\theta^2)$$ Now it has to be shown that $\theta_{ij}$ is real and anti-symmetric. I've started ...
4
votes
1answer
60 views

solvable subalgebra

I want to show that a set $B\subset L$ is a maximal solvable subalgebra. With $L = \mathscr{o}(8,F)$, $F$ and algebraically closed field, and $\operatorname{char}(F)=0$ and $$B= ...
2
votes
2answers
338 views

Weyl group of the Lie algebra $\mathfrak{sl}_n$

The Weyl group of the Lie algebra $\mathfrak{sl}_n$ is just the symmetric group on $n$ elements, $S_n$. The action can be realized as follows. If $\mathfrak{h}$ is the Cartan subalgebra of all ...