For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

learn more… | top users | synonyms (1)

12
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 ...
30
votes
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 ...
5
votes
4answers
167 views

Showing the Lie Algebras $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$ are not isomorphic.

I am working through the exercises in "Lie Groups, Lie Algebras, and Representations" - Hall and can't complete exercise 11 of chapter 3. My aim was to demonstrate that there does not exist a vector ...
4
votes
3answers
192 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 ...
3
votes
1answer
243 views

Radical of $\mathfrak{gl}_n$

I find it intuitive enough that the radical of $\mathfrak{gl}_n\mathbb F$ is the scalar matrices, but I have trouble finding an easy, but complete proof: Proof. Let $\mathfrak s$ denote the scalar ...
2
votes
2answers
195 views

Solvable equivalent to nilpotency of first derived Lie algebra?

The Wikipedia "Solvable Lie Algebra" page lists the following property as a notion equivalent to solvability: $\mathfrak{g}$ is solvable iff the first derived algebra $[\mathfrak{g},\,\mathfrak{g}]$ ...
8
votes
2answers
4k 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 ...
11
votes
6answers
715 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 ...
8
votes
4answers
365 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 ...
11
votes
2answers
827 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 ...
8
votes
2answers
1k 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 ...
4
votes
3answers
535 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, ...
6
votes
2answers
789 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 $[X,Y] = A$ ...
4
votes
2answers
409 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 ...
1
vote
2answers
158 views

How to “find” this Lie algebra: proof that $\mathfrak{sl}$ is trace zero matrices

I saw this table here on Wikipedia and it states that the Lie algebra of the special linear group $SL_n(\mathbb C)$ is the group of traceless matrices $\mathfrak{sl}_n$. I know the definition of a ...
3
votes
2answers
214 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 ...
1
vote
1answer
36 views

The center of a nilpotent Lie algebra intersects each ideal

If $\cal h$ is a nonzero ideal in a nilpotent Lie algebra $\cal g$. How to prove that $\mathcal h\cap Z(\mathcal g)\not =0$, where $Z(\mathcal g)$ is the center of $\mathcal g$?
0
votes
2answers
114 views

$ gl(2,\mathbb C) \cong sl(2,\mathbb C) \oplus \mathbb C $

Hi I just start learning Lie algebra and there is one hw question I don't really understand how to do, hope somebody give me some hints. $L_1,L_2$ are Lie algebras. $L=\{(x_1,x_2):x_i \in L_i\}$. Lie ...
11
votes
3answers
603 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
729 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
votes
0answers
130 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 ...
5
votes
2answers
427 views

Representations of Direct Sum of Lie Algebras

I'm trying to prove the following. Let $\frak{g}$ and $\frak{h}$ be (semisimple) Lie algebras. Then every representation $d$ of $\frak{g}\oplus\frak{h}$ is the tensor product of representations $d^1$ ...
8
votes
2answers
946 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
383 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 ...
2
votes
0answers
30 views

Tensor formula in SU(3) representations

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
2
votes
1answer
71 views

$\mathfrak{su}(2)$ not isomorphic to $\mathfrak{sl}(2,\mathbb R)$

I would like to prove that the two algebras $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb R)$ are not isomorphic as Lie algebras. I started considering the following basis for $\mathfrak{su}(2)$: ...
1
vote
1answer
56 views

Proof that these two definitions are equivalent

Where can I find a proof of or how can I prove that these two definitions are equivalent? Definition 1: The Lie algebra of a Lie group $G \subset GL_n$ is the tangent space at $I$. Definition 2: ...
10
votes
5answers
369 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 ...
2
votes
1answer
289 views

Does the abstract Jordan decomposition agree with the usual Jordan decomposition in a semisimple Lie subalgebra of endomorphisms?

Is it true that for every element $x$ of a semisimple Lie subalgebra of endomorphisms $L\subseteq \text{End}(V)$, where $V$ is a finite dimensional vector space over $\mathbb{C}$, the abstract Jordan ...
1
vote
1answer
55 views

Are the $C$-points of a simply connected algbraic group simply connected?

Let $G$ be a simply connected algebraic group defined over $C$. Note : I see the definition of simply connected as "Every isogeny to $G$ is an isomorphism" as given in Hochschild's "Basic Theory of ...
5
votes
2answers
254 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 ...
4
votes
2answers
223 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
149 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
196 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
180 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
1answer
162 views

Lie algebras and roots systems

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 ...
3
votes
2answers
586 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
85 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
35 views

Radical of a Lie algebra bracket itself

Let $\mathfrak{g}$ be a Lie Algebra over $k$, $\mathfrak{n}$ its radical. Why is $[\mathfrak{n},\mathfrak{g}]$ the smallest of its ideals $\mathfrak{a}$ such that $\mathfrak{g}/\mathfrak{a}$ is ...
1
vote
1answer
63 views

Let $\rho$ f.d. rep of a nilpotent Lie algebra such that $\rm{det} \rho(X) = 0$, $\forall X$. Then $\exists v \neq 0$: $\rho(X)v = 0, \forall X$.

Could you help me with this question? Let $\mathcal{g}$ be a nilpotent Lie algebra over $k$ and $\rho$ a representation of $\mathcal{g}$ in a finite-dimensional nonzero vector space $V$ over $k$. ...
1
vote
2answers
69 views

Computing information about a Lie algebra from cartan matrix

Let's consider the Cartan matrix : $$ \begin{pmatrix} 2 & -2 \\ -1 & 2 \end{pmatrix} $$ I am asked to find the number of roots and then to compute character of the adjoint representation. ...
1
vote
1answer
94 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
37 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 ...
1
vote
1answer
34 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
1answer
120 views

Borel subalgebras contain solvable radical

Let $L$ be a Lie algebra and let $B$ be a Borel subalgebra (a maximal solvable subalgebra) of $L$. I want to understand why $\operatorname{Rad} L \subseteq B$. In his proof, Humphreys ...
1
vote
2answers
133 views

If Lie algebra is equal to its commutant, is it semisimple?

I am searching for a prove or a counterexample for this statement: If finite-dimensional complex Lie algebra is equal to its commutant, then it is semisimple. I suppose it is not true, because ...
1
vote
2answers
399 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
42 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
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 $\mathfrak so(V,B)$ ...
0
votes
1answer
116 views

Is it true if $[LL] = L$ then $L$ is a semisimple Lie algebra?

Let $L$ be a finite dimensional Lie algebra over $\mathbb{C}$. It is classical theorem that if $L$ is semisimple, then $[LL] = L$. Is it true if $[LL] = L$ then $L$ is a semisimple Lie algebra? I've ...