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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28
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4answers
1k views

“Cayley's theorem” for Lie algebras?

Groups can be defined abstractly as sets with a binary operation satisfying certain identities, or concretely as a collection of permutations of a set. Cayley's theorem ensures that these two ...
8
votes
2answers
3k views

Classsifying 1- and 2- dimensional Algebras, up to Isomorphism

I am trying to find all 1- or 2- dimensional Lie Algebras "a" up to isomorphism. This is what I have so far: If a is 1-dimensional, then every vector (and therefore every tangent vector field) is of ...
8
votes
4answers
350 views

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty ...
10
votes
2answers
2k views

On surjectivity of exponential map for Lie groups

A recent question made me realize I didn't know any proof that exponential of a Lie algebra $\mathfrak g$ of a compact connected Lie group $G$ is surjective. After a bit of thinking I've come up with ...
9
votes
2answers
602 views

Which Lie groups have Lie algebras admitting an Ad-invariant inner product?

I am trying to answer the following question: Which Lie groups have a Lie algebra admitting an $\text{Ad}$-invariant inner product? First of all, all compact Lie groups satisfy this condition ...
4
votes
2answers
295 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 ...
4
votes
2answers
548 views

Is every skew-adjoint matrix a commutator of two self-adjoint matrices

I'm looking to solve some matrix equations. One of the equations involves a commutator, so my question is as follows: let $A$ be a skew-self-adjoint, traceless matrix, does the equation $[X,Y] = A$ ...
3
votes
2answers
201 views

Defining an isomorphism that respects the Lie bracket: is my work correct?

I previously determined that if $\mathfrak{sl}$ denotes the Lie algebra of $SL_2(\mathbb C)$ and $\mathfrak o$ denotes the Lie algebra of $O(3,\mathbb C)$ then a basis for $\mathfrak{sl}$ is given by ...
11
votes
3answers
400 views

Translations in two dimensions - Group theory

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
7
votes
1answer
574 views

Reference for Lie-algebra valued differential forms

I am learning about vector-valued differential forms, including forms taking values in a Lie algebra. On Wikipedia there is some explanation about these Lie algebra-valued forms, including the ...
5
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0answers
112 views

— Cartan matrix for an exotic type of Lie algebra --

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra? For example, consider this Lie algebra: $$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = - f_{ik}{}^j Y^k \qquad\qquad ...
8
votes
2answers
794 views

Universal Cover of $SL_{2}(\mathbb{R})$

Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite dimensional representations?
4
votes
1answer
346 views

Why Lie algebras of type $B_2$ and $C_2$ are isomorphic?

both of Lie algebras of type $B_2$ and $C_2$ have dimension 10 and we can find two basis of them on page 3 in the book: Introduction to Lie algebras and representation theory . How could we show that ...
10
votes
5answers
312 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
3answers
174 views

How to prove that $B^\vee$ is a base for coroots?

Let $\Phi$ be a root system in a real inner product space $E$. Define $\alpha^\vee = \frac{2\alpha}{(\alpha, \alpha)}$. Then the set $\Phi^\vee = \{\alpha^\vee: \alpha \in \Phi \}$ is also a root ...
2
votes
1answer
105 views

Proving $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$

Let $X,Y$ be vector fields. $L_X$ is the Lie derivative and $i_X$ is the contraction of a $k$-form. I am really stuck on how you could prove the identity $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$. Update: I ...
1
vote
1answer
29 views

Lie algebra of affine linear maps

Let $G$ be the Lie group of affine transformations, $$\{x \mapsto Ax+b,A \in GL(n), b \in \mathbb{R}^n\}.$$ We can represent these maps as matrices $$\begin{pmatrix} A & b \\ 0 & 1 ...
5
votes
2answers
235 views

Subgroups of $\Bbb{R}^n$ that are closed and discrete

I am trying to prove that every closed discrete subgroup of $\Bbb{R}^n$ under addition is a free abelian group of finite rank. I have tried to do this by induction on the dimension $n$. Base ...
3
votes
2answers
98 views

The generators of $SO(n)$ are antisymmetric, which means there are no diagonal generators and therefore rank zero for the Lie algebra?

Okay, this may be a silly question but I can't figure it out myself right now. By definition $O \in SO(n)$ fulfils $O^T O=1$ and $\det(O)=1$. For the generators of the group $ T_a \in so(n)$, this ...
3
votes
0answers
87 views

How to understand Weyl chambers?

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane ...
3
votes
1answer
128 views

Help needed in understanding the basics of Cartan decomposition of a Lie algebra

I am trying to learn the basics of Cartan decomposition of Lie algebra, and have come across the following example. Consider $\mathfrak{gl_n}$ as the Lie algebra of endomorphisms of $\mathbb{C}^n$. ...
3
votes
2answers
120 views

Is every element of a complex semisimple Lie algebra a commutator?

Let $L$ be a (finite-dimensional) complex semisimple Lie algebra. Then we know that $L = [L,L]$. Is it true that every element of $L$ must be a commutator? Since a complex semisimple Lie algebra is ...
3
votes
2answers
520 views

Lie Algebra Homomorphism Question

So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
2
votes
1answer
79 views

How to show trace of $AB$ is zero for $A \in \mathfrak{u}_n$ and $B \in \mathcal{H}_n$?

Please have a look at this question: Help needed in understanding the basics of Cartan decomposition of a Lie algebra I want to show that the decomposition $\mathfrak{gl}_n = u_n \oplus ...
1
vote
1answer
73 views

Where does the ambiguity in choosing a basis for a Lie algebra come from?

This is a follow-up to this question. For matrix Lie algebras, we can define the Lie algebra $g$ of a group $G$ as the set $T_a \in g$ that yield an element of $G$ when put into the exponential map: ...
1
vote
1answer
29 views

Solving a large system of linear equations to satisfy the Lie bracket: am I doing it right?

I'm still working on a Lie algebra isomorphism from the Lie algebra of $SL_2(\mathbb C)$ into the Lie algebra of $O(3, \mathbb C)$. It has been suggested to me to use linear combinations of ...
1
vote
2answers
307 views

A covering map from a differentiable manifold

Let $p: C \to X$ is a covering map. Suppose that $C$ is a differentiable manifold. Is X - differentiable manifold? More precisely, I am interested in the case where $C$ is Submanifold of Lie algebra, ...
1
vote
0answers
37 views

System of roots [duplicate]

Let $\Phi$ an irreducible system of roots, $\Phi^{+} \subset \Phi$ a choose of positive roots. I have to prove that if $(\alpha, \beta) \ge 0$ for al $\beta \in \Phi^{+}$ then $\alpha$ is the highest ...
0
votes
0answers
20 views

Finding coefficients for this Lie algebra isomorphism

This is a question closely related to my previous questions. How, in this thread here did Hee Kwon Lee find the coefficients $(-i,1,0),\ (-i,-1,0)$ and $(0,0,2i)$? In the linked thread ...
0
votes
0answers
29 views

$\mathfrak so(V,B)$ as subalgebra and trace if subsets of it.

I'm studying lie algebras, and got stuck on this one: Let $B$ be a bilinear form on a finite-dimensional vector space $V$ over $\mathbb F$. I've seen many books that say that $\mathfrak so(V,B)$ ...
6
votes
1answer
2k views

Question on fundamental weights and representations

I am a bit confused about the notion of "fundamental weights". In a complexified setting, I am thinking of my Lie algebra to be decomposed as, $\cal{g} = \cal{t} \oplus _\alpha \cal{g}_\alpha$ where ...
9
votes
3answers
3k views

How do you find the Lie algebra of a Lie group (in practice)?

Given a Lie group, how are you meant to find its Lie algebra? The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them? My group is the set ...
10
votes
5answers
505 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 ...
4
votes
2answers
535 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
2k views

Lie derivative of a vector field equals the lie bracket

Let $X$ and $Y$ be vector fields on a smooth manifold $M$, and let $\phi_t$ be the flow of $X$, i.e. $\frac{d}{dt} \phi_t(p) = X_p$. I am trying to prove the following formula: $\frac{d}{dt} ...
8
votes
2answers
640 views

Canonical isomorphism between $\mathfrak{so}(3)$ and $\mathbb R^3$ with vector cross product

There is a well-known isomorphism between the Lie algebra $\mathfrak{so}(3)$ and $\mathbb{R}^3$ which maps the Lie bracket to the vector cross product. It looks like $$ \begin{pmatrix} 0 & -z ...
7
votes
4answers
658 views

What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
15
votes
3answers
1k views

Lie algebra of a quotient of Lie groups

Suppose I have a Lie group $G$ and a closed normal subgroup $H$, both connected. Then I can form the quotient $G/H$, which is again a Lie group. On the other hand, the derivative of the embedding ...
10
votes
2answers
233 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
6
votes
2answers
988 views

symplectic lie algebra is simple

The symplectic lie algebra defined by $sp\left(n\right)=\left\{ X\in gl_{2n}\,|\, X^{t}J+JX=0\right\}$ when $J=\begin{pmatrix}0 & I\\ -I & 0\end{pmatrix}$. So $X\in sp\left(n\right)$ is of ...
5
votes
1answer
332 views

Definition of a “root” of a Lie Algebra

I am using the notation that $g$ is the Lie algebra of the Lie group $G$ and $T$ is the maximal torus of $G$ and $t$ is the Lie algebra of $T$ (and hence $t$ is the Cartan subalgebra of $g$). A ...
7
votes
1answer
316 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 ...
6
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3answers
749 views

Physical interpretation of the Lie Bracket

I've come accross this physical interpretation for $ [X,Y] $ which I don't understand : Follow $X$ for some time $\epsilon$; Follow $Y$ for $\epsilon$; Follow -X for $\epsilon$; Follow -Y for ...
5
votes
3answers
934 views

How should I show that the Lie algebra so(6) of SO(6) is isomorphic to the Lie algebra su(4) of SU(4)?

As far as I can see, an isomorphism of Lie algebras is a bijective map which preserves the Lie bracket. I need to show that $\mathfrak{so}(6)$ (the Lie algebra of SO(6)) is isomorphic to the ...
5
votes
1answer
760 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?
3
votes
2answers
76 views

Lie group step in proof

Let $X_e,Y_e \in T_eG$ be vectors and $G = GL(n).$ Then the right translation is given by $Y_g = Y_e g$ and $X_g = X_e g.$ Now, I have a proof showing that $[X_e,Y_e] \in T_eG$ is the element ...
2
votes
1answer
107 views

Question regarding isomorphisms in low rank Lie algebras

I am reading Brian Hall's book 'Lie Groups, Lie Algebras, & Representations' and on p.271 I find that in low rank Lie algberas there are some isomorphisms. For example, ...
2
votes
1answer
572 views

How to draw a weight diagram?

Given a weight, say $\omega=3\lambda_1+4\lambda_2$, where $\lambda_1, \lambda_2$ are fundamental weights (type A Lie algebra). How to draw the weight diagram of the irriducible representation with ...
7
votes
1answer
171 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 ...
7
votes
1answer
210 views

Three-dimensional simple Lie algebras over the rationals

Let $\mathfrak g$ be a three-dimensional $\mathbf Q$-vectorspace endowed with the structure of a simple Lie algebra. How many non-isomorphic such $\mathfrak g$ are there? Over the complex numbers, ...