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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3
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0answers
43 views

Would the transformation of a differential equation obey the same algebra?

I've found that the algebra of this differential equation $$\frac{d^2y}{dz^2}-(3z^2+\gamma)\frac{dy}{dz}+(cz+\alpha)y=0$$ is in $sl(2)$ because it is possible to use the generators of the $sl(2)$ ...
2
votes
1answer
41 views

Complete reducibility of a field extension of an lie algebra representation

Let $\mathfrak{g}$ be a lie algebra over a field $k$ with characterstic $0$ and $k\subset k'$ a finite field extension. Suppose $\mathfrak{g}\otimes k'$ has the property, that all finite dimensional ...
2
votes
1answer
36 views

Question on Left-Invariant Vector Fields

Let $G$ be a Lie group, and $\xi \in T_{e}G$ a tangent vector at the identity. Given a function $f \in C^{\infty}(G)$, verify that $ g \rightarrow ((\ell_{g})_{*}\xi)f$ is a $C^{\infty}$ function on ...
2
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0answers
41 views

Fulton-Harris Lemma 3.35

In the proof of Lemma 3.35 in Fulton--Harris, Representation Theory, it is claimed that the identification $H(\phi^2(x),y)=H(x, \phi^2(y))$ implies that $\lambda$ is a positive real ($\phi^2$ is known ...
1
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0answers
32 views

Explicit Weyl group invariant polynomials

Quoting this post, "Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset\mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to g. Let ...
0
votes
1answer
42 views

Kernel of homomorphism on unit circle S1

Let $f : S^1 \to S^1$ be defined such that $f(z) = z^2$, where $z$ is a complex number. It's easy to check that this is a homomorphism on $S^1$. However, how would you find the kernel and the coset ...
1
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1answer
21 views

Weyl function defintion

In Lie algebra book by Humphreys, in the section 24.1 (page number 136) he defines Weyl function $q$ as $ q = \Pi_{\alpha \gt 0}(\epsilon_{\frac{\alpha}{2}}-\epsilon_{-\frac{\alpha}{2}})$. I can't ...
2
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0answers
31 views

Nilpotency of Lie algebra from structure constants

Consider a given set of structure constants $c_{ij}^k$ defining a (finite dimensional) Lie algebra $\mathfrak{L}$, i.e. $$[e_i,e_j] = \sum_{k=1}^N c_{i,j}^k \, e_k \qquad i,j=1,\ldots,N$$ with $N$ ...
2
votes
1answer
27 views

Auto-Langlands dual gruops.

Consider a semisimple Lie group $G$. We define the Langlands dual $\hat{G}$ of $G$ as the group which has as a root system, the root system generated by the coroots of $G$. Recall that given a root ...
1
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1answer
25 views

Weyl group and weight lattice chambers.

Consider two simple Lie groups $G_1$ and $G_2$. Let $G_1$ have $W_1$ as a Weyl group and $G_2$ have $W_2$ as a Weyl group. Is it true that the Weyl group of $G_1 \times G_2$ is $W_1 \times W_2$? ...
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0answers
33 views

Reference request: exact sequences of Lie algebras

I have a reference request: where can I read more about the following? Consider the short exact sequence $0\rightarrow \mathfrak{n}^- \rightarrow \mathfrak{gl}_n\rightarrow \mathfrak{b}\rightarrow ...
1
vote
2answers
43 views

How to construct the Lie bracket from a Lie group?

Suppose you have a Lie group $G$ with identity element $g$, then the Lie algebra is isomorphic to the tangent space at $g$, $T_gG$. However, to fully specify the Lie algebra, you also have to define a ...
2
votes
2answers
48 views

Semisimple Lie algebras are perfect.

Can anyone explain why a semi-simple finite dimensional Lie algebra $\mathfrak{g}$ has to be perfect ? The natural way to prove something like that would be to look to the algebra generated by the ...
1
vote
0answers
36 views

Which is the Weyl group of $U(n)$

Consider the unitary group $U(n)$. How does one compute its Weyl group? Is it the same as the Weyl group of $SU(n)$ since $U(n)\simeq SU(n)\times U(1)$?
10
votes
1answer
117 views

Is it really unknown that every endomorphism of the Weyl algebra $A_1$ is an isomorphism?

Here $A_1 := K\{x\cdot-, \frac{d}{dx}\} \subset \operatorname{End}_K(K[x])$ for some characteristic-zero field $K$. I found this claim in Coutinho's "A Primer of Algebraic D-Modules." If this is ...
0
votes
0answers
14 views

Invariants of exterior power of Lie algebras

Let $\mathfrak{g}$ a simple finite dimensional Lie algebra, and consider $$\bigwedge(\mathfrak{g}\oplus\mathfrak{g}).$$ Let $\{e_i\}$ and $\{f_i\}$ be dual basis of $\mathfrak{g}$ with respect to the ...
0
votes
0answers
19 views

Invariants of representation of simple Lie algebras.

Let $\mathfrak{g}$ a finite dimensional simple Lie algebras and let $V$ a representation of $\mathfrak{g}$ such that $$V=\bigoplus_{i,j\in I}(L(\mu_i)\otimes L(\mu_j)).$$ Where $L(\mu_i)$ is the ...
1
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1answer
54 views

Reference for Weyl Character Formula

I am reading Lie algebra book by James E.Humphreys. This book giving enough discussion about Weyl Character Formula and its proof, still I would like to know what are the other books or lecture notes ...
0
votes
0answers
35 views

holomorphic Lie group automorphisms of the complex Heisenberg group preserving the lattice

I don't understand Lie group,but I need an elementary result about it,so I ask for help for a simple question. Take a complex Heisenberg group $G$ $$ \left( \begin{array}{ccc} 1 & z_1 & ...
1
vote
1answer
48 views

Models for Lie algebra E8 and octonions

I've heard that one can construct the exceptional Lie algebra $E_8$ as the Lie algebra of the group of isometries of projective plane over octonions, or something of this form. Unfortunately, I do not ...
3
votes
1answer
25 views

Does the exponential map respect module actions?

Setup: Let $k$ be a field and $G \subseteq \mathrm{GL}_n(k)$ an algebraic group, reductive if that makes a difference. Let $\mathfrak g \subseteq \mathfrak{gl}_n(k)$ be the Lie algebra of $G$ with ...
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0answers
39 views

Exercise in Lie algebra course

Let $A$ and $B$ be subalgebras of a Lie algebra $L$ such that $B\subset N_L(A)$. (a) Verify that the space $A+B$ is a subalgebra of $L$. (b) Verify that $A\triangleleft(A+B)$ and $(A\cap ...
2
votes
0answers
20 views

intuition behind basis of a root system

I am reading Lie Algebra book by James E.Humphreys. I can understand the fruitfulness of the notion of basis of a root system. But what is the intuition behind this definition, In particular the ...
1
vote
1answer
62 views

Questions on Killing form: its definition and a root space decomposition.

I have a question on Killing form. Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Consider the adjoint representation $(\mathrm{ad},\mathfrak{g})$ of $\mathfrak g$, i.e. $$ \mathrm{ad}: ...
0
votes
0answers
33 views

Irrep dimensions of non semisimple Lie algebra

I'm mostly interested in Lie algebra "numerology". The book "Birdtracks" and the website http://www-math.univ-poitiers.fr/~maavl/LiE/form.html answered me everything on irrep dimensions for semisimple ...
0
votes
0answers
22 views

Constructing Lie algebra from the associative algebra.

Show that any associative algebra $A$ can be made into Lie algebra by taking $[x,y]=xy-yx$ for any $x,y \in A$. The way I would tackle it. $\circ$ Clearly $A$ is a vector space as it is an ...
0
votes
0answers
29 views

Heisenberg algebras and their ideals

We know that Heisenberg Lie algebra is a Lie algebra $H(2m+1)$ with basis $v_1, \ldots , v_{2m}, v$ and the only non--zero multiplication between basis elements is given by $[v_{2i-1}, v_{2i}] = - ...
0
votes
2answers
74 views

What are the good textbooks on Kac-Moody groups?

While there is a number of good books on Kac-Moody algebras ("Infinite dimensional Lie algebras" by Kac is already enough), it seems to me there is lack of textbooks on Kac-Moody groups. nLab says ...
1
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1answer
63 views

Question about Cartan involution on wikipedia

I am a beginner in representation theory. I have some questions about Cartan involution. The following is the link in wikipedia http://en.wikipedia.org/wiki/Cartan_involution My question is about ...
0
votes
1answer
54 views

Infinite number of Lie groups with the same lie algebra

Is there a finite dimensional Lie algebra L such that there are infinite number of non isomorphic compact connected lie groups which Lie algebras are isomorphic to L?
3
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0answers
59 views

Invariant subalgebra of a Lie algebra under an automorphism of the Dynkin diagram

Of all the automorphisms of a (finite-dimensional, semisimple) Lie algebra which induce a particular automorphism of its Dynkin diagram, is there a particular one which is "nicer" than the others? ...
0
votes
1answer
25 views

Given the generators of a group find the parametrization matrix

I have the generators of $sl(2,\mathbb{R})$ algebra $$J_0=\begin{pmatrix}0&1\\ -1&0\end{pmatrix},\quad J_1=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix},\quad J_2=\begin{pmatrix}0&1\\ ...
1
vote
1answer
49 views

Is it true for solving differential equations by getting constant coefficient matrix with magnus expansion

The magnus expansion is given in detail http://en.wikipedia.org/wiki/Magnus_expansion. While implementing magnus expansion to differential equations we have an iteration formula as follows $$Y'(t) = ...
1
vote
1answer
30 views

Why is that an automorphism that preserves $B$ and $H$ an automorphism of $\Phi$ that leaves $\Delta$ invariant?

Let $L$ be a semisimple finite dimensional Lie algebra, $H$ its CSA and $\Phi$ its root system with base $\Delta$ and $B = B(\Delta) = H\bigoplus_{\alpha \succ 0}L_\alpha$. If we have an automorphism ...
9
votes
1answer
105 views

Category of Lie group representations equivalent to the category of representations of their Lie algebra

Let $G$ be a lie group and $\mathfrak{g}$ its lie algebra. Consider the category $Rep(G)$ of finite dimensional representations of $G$ and the category $Rep(\mathfrak{g})$ of finite dimensional ...
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0answers
27 views

Simple question: Lie algebra and p-groups

Assume $p$ is a prime and $\pi$ is the set of primes dividing $(p-1)!$. $\mathbb{Q}_{\pi}$ is the set of all rational numbers with $\pi$-numbers as denominators. A $\pi$-number is a product of ...
1
vote
1answer
75 views

How do I know that a group generator really is from that group?

I've asked the question here but I'm actually not that satisfied with an answer. I have 3 generators that are (at least from what I read in every article I find) generators of $SL(2,\mathbb{R})$ ...
1
vote
1answer
38 views

The set of ad-nilpotent elements in an algebra

Let $N\subset L$ denote the set of ad-nilpotent elements in a Lie algebra $L$. I verified that $N$ is a subspace of $L$, but it seems to me that $N$ might not be a subalgebra, Let $x$ and $y$ be ...
1
vote
1answer
47 views

How does a semisimple Lie algebra determine its root space?

I understand that given a root system $\Phi$, by Serre's theorem there exists a Lie algebra $L$ with root system $\Phi$. Also isomorphism theorem implies that any two such $L$ are isomorphic. That ...
3
votes
1answer
59 views

Precise connection between complexification of $\mathfrak{su}(2)$, $\mathfrak{so}(1,3)$ and $\mathfrak{sl}(2, \mathbb{C})$

I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the ...
2
votes
1answer
53 views

Lie algebra of $\Bbb{R}^n$

I don't really understand the Lie algebra of the Euclidean space (the bold part): Left translation by an element $b\in\Bbb{R}^n$ is given by the affine map $L_b(x)=b+x$, whose pushforward $(L_b)_*$is ...
0
votes
1answer
44 views

Simple Lie algebras have irreducible root systems?

I was unable to see why $(\alpha+\beta,\alpha) \ne 0$ and $(\alpha+\beta,\beta)\ne0$ implies $\alpha+\beta \not\in\Phi$. Everything else is fine. $\quad$*Proposition.* Let $L$ be a simple Lie ...
3
votes
1answer
50 views

— Cartan matrix for a semisimple Lie algebra with an extension

The question is a modified one inspired by this post: What is the Cartan matrix for this Lie algebra below? (for this semisimple Lie algebra $g(X) \oplus h(Y)$,) $$ [X_i, X_j] = f_{ij}{}^k X_k ...
2
votes
1answer
51 views

Reducing size of ODE system by using symmetries: examples, references help request.

We know: A high order differential equation can be expressed as an ODE system. Knowledge of a symmetry allow one to reduce the order of a differential equation. So if we do $n$-order ODE ...
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0answers
38 views

Questions about affine Weyl group and extended affine Weyl group for SL2.

Let $G=SL_2$. Then the Weyl group is generated by $s_1$. On page 3 of the lecture notes, it is said that the affine Weyl group is generated by $s_0, s_1$. (1) The element $s_0s_1$ can be identified ...
0
votes
1answer
17 views

Killing form for a non-abelian Lie Algebra of dimension $2$

Aratati ca forma Killing pentru algebra Lie ne-abeliana de dimensiune 2 nu este zero. How can I prove that the Killing forme of a non-abelian Lie algebra is not equal with $0$? Thanks
-1
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1answer
32 views

About Killing Form [closed]

How can I prove that why the Killing form for an abelian Lie algebra is $0$? Can you help with suggestions, references, answers? thanks!
2
votes
1answer
51 views

Regarding root space decomposition

In Humphreys, given a finite dimensional semisimple Lie algebra $L$ and a maximal toral subalgebra $H$, $$L_\alpha := \{x\in L|[hx] = \alpha(h)x\;\forall h\in H\}$$ Then since $ad_L\;H$ is a commuting ...
6
votes
3answers
188 views

Prove where exp: Skew($3\times 3$) $\rightarrow SO(3)$ is local homeomorphism

The matrix exponential on skew-symmetric $3\times3$ matrices onto $SO(3)$ is not local homeomorphism everywhere. I have been instructed that one problem is with the spheres of radius $2n\pi$ ...
5
votes
1answer
107 views

Whether matrix exponential from skew-symmetric 3x3 matrices to SO(3) is local homeomorphism?

$SO(3)$ denotes 3x3 rotation matrices. This is Lie group, with corresponding Lie algebra being $\mathrm{Skew}_3$, the space of 3x3 skew-symmetric matrices. The link between them is the matrix ...