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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4
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1answer
66 views

Epimorpsims preserve generalized eigenspaces

This is most likely trivial, but I don't get it. In Humphrey's Introduction to Lie algebras, page 82, he says: It is clear that, if $\phi \colon L \to L'$ is epimorphism [of finite dim. Lie ...
1
vote
0answers
118 views

Dimension of the root spaces of a semisimple complex Lie algebra

I have problems in understanding the proof that the root spaces of a semisimple Lie algebra are all 1-dimensional and that the only multiples of a root $\alpha \in \Phi$ which occur in $\Phi$ are $\pm ...
3
votes
2answers
314 views

Under what conditions is the exponential map on a Lie algebra injective?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\exp :\mathfrak{g}\rightarrow G$ be the exponential map. In his blog, Terrence Tao notes that if a Lie group is not simply-connected, ...
2
votes
1answer
108 views

Problem with the definition of simple Lie algebra

I have the following definitions: Simple Lie algebra: A Lie algebra $\mathfrak{g}$ is simple if it ha no non-trivial (i.e. $0$ or $\mathfrak{g}$ itself) ideals. Semisimple Lie algebra: A Lie algebra ...
4
votes
4answers
1k views

Could you recommend some books on Lie algebra?

I am a pure maths student, and want to go straight ahead, so I decide to study Lie algebra on my own, and try my best to understand it from various points of view:differential equation, Lie group, ...
7
votes
1answer
163 views

Ideal defining the nilpotent cone of $\mathfrak{gl}_n(k)$

Let $k$ be an algebraically closed field, and let $\mathfrak{g}=\mathfrak{gl}_n(k)$. Let $\mathcal{N}\subset\mathfrak{g}$ be the nilpotent cone, that is: $$\mathcal{N}=\{A\in\mathfrak{g}\mid ...
1
vote
0answers
89 views

Cartan subalgebra

Let $g$ be a real semisimple Lie algebra with Cartan decoposition $(l,p)$. How can we show that a Cartan subspace $a$ of $p$ (Cartan subspace of $p =$ maximal element in a set that consists of all Lie ...
5
votes
1answer
324 views

Lie algebra of a Normalizer

Let $G$ be a Lie group and let $\mathfrak{g}$ be its Lie algbera. Let $E$ be a subspace of $\mathfrak{g}$. Define the normalizer of $E$ in $G$ as: $$ N_{G}(E) = \{ g \in G \;\; | \;\; Ad(g)E = E\}$$ ...
1
vote
1answer
54 views

$x$ is a nilpotent endomorphism implies $\operatorname{ad}x$ nilpotent

Lemma: Let $x\in\mathfrak{gl}(V)$ be a nilpotent endomorphism. Then $\operatorname{ad}x$ is also nilpotent. Proof: We may associate to $x$ two endomorphisms of $\operatorname{End}V$, left and ...
1
vote
1answer
42 views

Solvability theorems

Let $L$ be a Lie algebra. (a) If $L$ is solvable, then so are all subalgebras and homomorphic images of $L$. (b) If $I$ is a solvable ideal of $L$ such that $L/I$ is solvable, then $L$ ...
3
votes
1answer
77 views

Automorphism in the special linear algebra $\mathfrak{sl}_2(F)$

If $L=\mathfrak{sl}(n,F), g\in GL(n,F)$, prove that the map of $L$ to itself defined by $x\rightarrow -gx^tg^{-1}$ ($x^t=$transpose of $x$) belongs to $\operatorname{Aut}L$. When $n=2,g=$identity ...
0
votes
1answer
57 views

Centre of a connected Lie group.

I want to show that if $G$ is a connected Lie group, then its centre, $Z(G)$ equals $Ker \ Ad$, where Ad is the adjoint function define by: $$\forall X \in G, Y \in \mathfrak{g}: Ad(X)(Y) = ...
2
votes
1answer
92 views

Irreducible representation and reductive Lie algebra

If we have complex reductive Lie algebra L and her finite dimensional representation $\phi$. How can we show that $\phi$ is irreducible iff restriction $\phi|_{[L,L]}$ is irreducible?
11
votes
3answers
379 views

What is the main use of Lie brackets in the Lie algebra of a Lie group?

I am beginner in Lie group theory, and I can't find the answer a question I am asking myself : I know that the Lie algebra $\mathfrak g$ of a Lie group $G$ is more or less the tangent vector of $G$ at ...
2
votes
0answers
111 views

$\mathfrak{sl}(3,F)$ is simple

Prove that $\mathfrak{sl}(3,F)$ is simple, unless $\operatorname{char}F=3$. [Use the standard basis $h_1,h_2,e_{ij}(i\neq j)$. If $I\ne 0$ is an ideal, then $I$ is the direct sum of eigenspaces for ...
1
vote
1answer
159 views

Inner automorphism of $\mathfrak{sl}(2,k)$, $\operatorname{char}(k)=0$ and adjoint action

If we have $sl(2,k)$, char $k = 0$, with standard basis $(x,y,h)$ and inner automorphism $\sigma = \exp(\operatorname{ad}x)\exp(-\operatorname{ad}y)\exp(\operatorname{ad}x)$. How can we show that ...
3
votes
1answer
70 views

Lie algebra of dimension $3$ and $L=[L,L]$ must be simple

Suppose $\dim L=3$ and $L=[L,L]$. Prove that $L$ must be simple. [Observe first that any homomorphic image of $L$ also equals its derived algebra.] Recover the simplicity of $\mathfrak{sl}(2,F)$, ...
5
votes
1answer
264 views

Center of $\mathfrak{sl}(n,F)$

Prove that $\mathfrak{sl}(n,F)$ (matrices with trace zero) has center $0$, unless $\operatorname{char}F$ divides $n$, in which case the center is $\mathfrak{s}(n,F)$ (scalar multiples of the ...
10
votes
5answers
270 views

Does non-commuting $\mathfrak{g}$ imply non-abelian $G$?

Question 8.1 in Kristopher Tapp's introductory text on matrix groups asks to show that $SO(n)$ is non-abelian ($n>2$) by finding two elements of $so(n)$ that do not commute. Why is this method ...
4
votes
1answer
183 views

Clifford Algebra for understanding Atiyah Singer Index Theorem Reference Request

I am interested in studying Atiyah Singer Index Theorem and Spin Geometry and would like to study Clifford Algebras and their representations for this purpose. I have a book 'Clifford Algebras : An ...
0
votes
1answer
42 views

Scalar multiplication of automorphism

Let $\phi$ be an automorphism from a Lie algebra $L$ to itself. Is it necessary that $\phi(-x)=-\phi(x)$ for an element $x\in L$. The automorphism must satisfy $[\phi(x),\phi(y)]=\phi([x,y])$. We ...
1
vote
1answer
45 views

Inner automorphisms as normal subgroup

An automorphism of the form $\exp(\operatorname{ad}x)$, $\operatorname{ad}x$ nilpotent, is called inner; more generally, the subgroup of $\operatorname{Aut}L$ generated by these is denoted ...
0
votes
1answer
65 views

Endomorphisms with trace zero

If $L=\mathfrak{gl}(V)$ or $L=\mathfrak{sl}(V)$, and if $g\in GL(V)$ is any invertible endomorphism of $V$, show that $gLg^{-1}=L$. Suppose $L=\mathfrak{gl}(V)$, and let $l\in L$. Clearly, ...
5
votes
1answer
154 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
113 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
116 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
74 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 ...
9
votes
5answers
433 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
225 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
287 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
156 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
342 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
79 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
75 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
57 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
81 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
228 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
121 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
276 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
96 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
664 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?