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)

1
vote
0answers
32 views

Integration on associated vector bundle

Let $G$ be a compact lie group and $\mathfrak{g}$ be its Lie algebra then we can construct the integral on $G\times \mathfrak{g}$ by $$\int_G\int_{\mathfrak{g}}f(x,Y)dxdY$$ Where $x\in G$ and $Y\in ...
1
vote
1answer
19 views

Help with proof in Humphreys (2)

Lemma: If $\mathfrak{k}$ is a subalgebra of $\mathfrak{g}$ that contains an Engel subalgebra, then $\mathfrak{k}$ is self-normalizing. Proof: Suppose $\mathfrak{k}\supset \mathfrak{g}_0(ad\; x)$ for ...
1
vote
1answer
21 views

Is the radical of a Lie algebra preserved by any of its derivations?

Let $\mathfrak{g}$ be a finite dimensional complex Lie algebra. A derivation $D: \mathfrak{g} \rightarrow \mathfrak{g}$ is $\mathbb{C}$-linear map $\mathfrak{g} \rightarrow \mathfrak{g}$ such that for ...
1
vote
1answer
57 views

How to find Casimir Operators and their degree.

Consider the quite general problem of computing all Casimir Operators of a given Lie Algebra $\mathfrak{g}$. How does one proceed, in general? And how is possible to compute the degree of a given ...
0
votes
0answers
26 views

Why $\widehat{G^{\mathbb{C}}}$ can be identified with the space of highest weights

Let $G$ be a compact connected Lie group and $G^{\mathbb{C}}$ be the complexification of Lie group $G$ and we denote $\widehat{G^{\mathbb{C}}}$ the set of isomorphism classes of irreducible rational ...
0
votes
1answer
20 views

Show that $(X^*)^*=\epsilon\epsilon 'X$

A finite dimensional vector space $V$ with a non-degenerate form (,) s.t. $(u,v)=\epsilon (v,u) \forall u,v\in V$ is called a quadratic space of type $\epsilon$. Let $V$ be a quadratic space of type ...
1
vote
1answer
36 views

Why $X^* X\in \mathfrak{g}(V)$.

A finite dimensional vector space $V$ with a non-degenerate form (,) s.t. $(u,v)=\epsilon (v,u) \forall u,v\in V$ is called a quadratic space of type $\epsilon$. Let $V$ be a quadratic space of type ...
0
votes
1answer
64 views

Tangent space of $\mathfrak{ so}(3)$ Lie algebra

Very basic question and the terminology makes it difficult to find a reference. I just know the basics of differential geometry but my question is simple. Is the tangent space at the point ...
1
vote
0answers
63 views

Structure constants of Lie algebra from system of linear equations

The Jacobi identity in terms of the structure constants $c_{p,q}^r$ of an $N$-dimensional Lie algebra with $p,q,r=1,\ldots,N$ reads $$ J_{i,j,k}^l \equiv c_{i,j}^m \, c_{k,m}^l + ...
0
votes
1answer
68 views

A question about differential forms on Lie groups

Let $G$ is a Lie group and $\mathfrak{g}$ is its Lie algebra and $\mathfrak{g}^{\mathbb{C}}$ be its complexification. Also assume that $\mathfrak{h}\subset \mathfrak{g}^{\mathbb{C}}$ be its ...
3
votes
1answer
93 views

Weyl Character Formula to find $M_\lambda(\mu)$

In Introductory Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the ...
0
votes
0answers
29 views

Weyl Character Formula to find $M_\lambda(\mu)$

In Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the dimensions of the ...
1
vote
2answers
50 views

Simple Lie algebra is a matrix algebra?

Wedderburn's Theorem. Let $A$ be a simple finite $k$-algebra. Then $A$ is a matrix algebra over a finite $k$-algebra $K$ which is a skew field. (Here matrix algebra means $A=M_n(K)$ for some $n$.) ...
0
votes
0answers
36 views

Equivalent representations of $\mathfrak{sl}_2$

Hello I have a question about the equivalence of two representations of the Lie algebra $\mathfrak{sl}_2$. The first representation is $(ad,\mathfrak{sl}_2)$ the adjoint representation with map ...
2
votes
0answers
34 views

A question in the proof that the weight of a finite dimensional module is W-invariant

Recently I'm reading Humphrey's book "Introduction to Lie algebra and representation theory", section 21 on the finite dimensional module of a semisimple lie algebra, and I have a question here which ...
2
votes
0answers
74 views

Invariant bilinear forms on Lie algebras

Consider a (compact) Lie group $H$ that acts on its Lie algebra $\mathfrak h$ in the usual way, $x\mapsto gxg^{-1}$ for any $x\in\mathfrak h$ and $g\in H$. Suppose we are given a real symmetric ...
1
vote
1answer
132 views

SO(2) group generator Lie Algebra

For the $2 \times 2$ orthogonal group of matrices which for the $SO(2)$ group, there is only one free parameter in the group element and hence only one generator for the group. Which is, $$ X_g = ...
1
vote
0answers
18 views

Show that $\langle[U,X],V\rangle = -\langle U,[V,X]\rangle$ for bi-invariant metric in Lie group

I know that $\langle U,V \rangle = \langle dR_{x_{t(e)}}U, dR_{x_{t(e)}}V \rangle$ and $\langle U,V \rangle = \langle dL_{x_{t(e)}}U, dL_{x_{t(e)}}V \rangle$ because it is bi-invariant. How do I ...
5
votes
3answers
110 views

Is the center of the universal enveloping algebra generated by the center of the lie algebra?

Let $\mathfrak{g}$ be a Lie algebra over a field $k$, and let $U(\mathfrak{g})$ be its universal enveloping algebra. $\mathfrak{g}$ is canonically embedded in $U(\mathfrak{g})$; identify it with its ...
6
votes
2answers
114 views

Equivalence of Two Lorentz Groups

How can I prove that $O(3;1)$ and $O(1;3)$ are the same group?
1
vote
1answer
47 views

Dimension of the weight space in stadard cyclic L-modules

Let $\lambda \in H^*$ be the irreducible standard cyclic module $V(\lambda)$ of weight $\lambda$ of a semisimple Lie algebra $L$. What are all the possible ways to determine : 1) Which $V(\lambda)$ ...
0
votes
1answer
64 views

Adjoint representation of $SL(2,\mathbb{R})$

Let $G$ be a Lie group. The adjoint representation $\text{Ad}:G \rightarrow \text{GL}(T_e G)$is given by $$ \text{Ad}(x):=T_e \mathcal{C}_x, $$ where $\mathcal{C}_x(y)=xyx^{-1}$. Now suppose that ...
2
votes
1answer
111 views

Generators of Translation - Lie Algebra [duplicate]

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} ...
10
votes
3answers
245 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} ...
1
vote
1answer
79 views

intuitive interpretation of Lie algebra

As you know, the isomorphism between $SO(2)$ and $e^{i\theta}$ allows an intuitive visualization of the Lie algebra $\mathfrak{so}(2)$ as the line $ti$. I wanted to know if there was a similar ...
5
votes
2answers
111 views

How does the lie algebra capture compactness of the lie group?

This is a soft question. Let $V,\rho$ be a representation of the lie algebra $\mathfrak{so}_3(\mathbb{R})$. Then if I understand everything right, $V$ is necessarily completely reducible, because the ...
3
votes
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
45 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
49 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
44 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
vote
0answers
53 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
57 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
22 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
40 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
49 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
37 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
36 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
48 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
62 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
54 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
126 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
17 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
vote
1answer
62 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
39 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
54 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
30 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
49 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
74 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}: ...