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)

3
votes
0answers
46 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
59 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
104 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
votes
0answers
55 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 ...
0
votes
1answer
122 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
vote
1answer
27 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
votes
0answers
52 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
139 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
vote
1answer
49 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 $W_1 \times W_2$? ...
1
vote
0answers
47 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 ...
2
votes
2answers
87 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
135 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
105 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
145 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 ...
1
vote
1answer
105 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 ...
2
votes
1answer
87 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
36 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 ...
1
vote
0answers
52 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
34 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
128 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
2answers
222 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
vote
1answer
227 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
59 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?
5
votes
1answer
220 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
34 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
69 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
38 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
191 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 ...
1
vote
0answers
29 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
82 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
64 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 ...
5
votes
1answer
312 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 ...
3
votes
1answer
79 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
72 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
72 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
98 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 ...
1
vote
0answers
173 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
23 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
votes
1answer
48 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
79 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
239 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
274 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 ...
5
votes
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 ...
2
votes
1answer
64 views

Reading help, ideal of an ideal is an ideal?

While in general it is not true that an ideal of an ideal is an ideal, this proof in Humphreys confuses me. Could someone please explain to me why the underlined sentences are true?
0
votes
1answer
51 views

How to build a basis for a vector space E(n+1) from a set of points given in E(n) (a vector space of rank n).

I'm interested in how (and if) one can build a new dimension from a set of given dimensions. Specifically, if we are given a vector space E(n) of rank n, and a sample S of elements of E(n) (let us ...
0
votes
1answer
57 views

Help with a proof in Humphreys

I was reading a proof but failed to see how the underlined step goes. How come ad $y = r($ad $s)$? That means $$r(a_i-a_j)e_{ij} = f(a_i-a_j)e_{ij} =ad\;y\;(e_{ij}) = r(ad\; s\;(e_{ij})) =r ...
3
votes
0answers
92 views

How was this Lie algebra found?

In a paper the author lists, without justification, generators for a Lie algebra. I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself. ...
1
vote
1answer
75 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 ...
2
votes
1answer
65 views

Humphreys proof check

Theorem: Let $L$ be a subalgebra of $\mathfrak{gl}(V)$ with $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V\ne 0$, then $\exists v\in V. v\ne 0$, such that $Lv = 0$. ...
0
votes
1answer
114 views

Characterisation of Cartan subalgebras as maximal toral; redundant “abelian” in the definition of “toral”

Edit: The reasoning on which this question is based is wrong. See my own answer for a counterexample. Let $L$ be a (finite dimensional) semisimple Lie algebra over a field $k$ with $char(k) = 0$. ...