1
vote
0answers
16 views

Lie algabra of R^n

Until now the only example of lie groups I have seen are subgroups of $GL_n$. Today I had the idea, that also $G=(\mathbb R^n,+)$ must be a lie group ($(\mathbb R^n,+)$ is a group with the ...
0
votes
1answer
31 views

In which course one learns Lie Group&Algebra and to which category of mathematics this subject belongs?

I'm a junior and i have never leanred this subject. I think "Lie Group&Algebra" is really deep and massive theory since the wikipedia page for it is quite long. Nevertheless, i'm not sure ...
1
vote
0answers
11 views

How could we decompose anticommutator of representation matrices for a Lie algebra?

For commutator, we know that $[T^a,T^b]=if^{abc}T^c$, where $f^{abc}$ is the structure constant. But is there a similar formula for $\{T^a, T^b\}$? Thank you.
3
votes
0answers
48 views

What is the kernel of a Maurer-Cartan form?

The Maurer-Cartan form on the Lie group $Gl(n,\mathbb{R})$ is a one-form taking values in $\mathfrak{gl}(n,\mathbb{R})$ as defined in the link. It has a rather concrete "extrinsic definition" as ...
2
votes
0answers
33 views

Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
1
vote
2answers
57 views

Why does exponentiating the derivative yield the shift operator?

If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
4
votes
1answer
23 views

If the Lie algebra is a direct sum, then the Lie group is a direct product?

I am reading the corollary 21.6 in the book "Morse Theory" by John Milnor, but I've encountered a statement for which I have no ideas. Let $G$ be a simply connected Lie group with a bi-invariant ...
2
votes
1answer
44 views

Representation theory and particle physics

Are there good books which explain clearly explain the connections between modern particle physics and representation theory of groups and lie algebras?
0
votes
0answers
16 views

Cartan subalgebra of product

i have a simple question what is the Cartan subalgebra of Lie algebra associated to the Lie group ?
1
vote
1answer
33 views

Advice on proving a tricky inequality

Im a little out of my depth here and am not well versed in combinatorics. Im not sure if this problem is too hard to solve or if there exists well known results to prove it. Here is part 1 which might ...
1
vote
2answers
43 views

Invariance under conjugation, equivalent in Lie Group and Lie Algebra?

Is the following true? $ e^X Y e^{-X} = Y \Leftrightarrow [X,Y]=0$ . From right to left you can show it with a corollary from the Baker–Campbell–Hausdorff formula. But in the other direction? I ...
1
vote
2answers
38 views

The discription of abelian Lie groups

There is a problem in my problem sheet to discribe all abelian connected Lie groups (moreover this is the first problem and it should be rahter easy). First it is difficult to understand how this ...
0
votes
0answers
16 views

Combining infinitesimal generators of diferent dimensions

I am reading a paper about ways in which you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of $SU(3)\times{}SU(2)\times{}U(1)$. At a certain point, it starts considering ways of getting ...
0
votes
1answer
19 views

A question on the nilradical and the nilpotent ideal of $\mathfrak{p}\subseteq \mathfrak{gl}_n(\mathbb{C})$

Let $\mathfrak{p}\subseteq \mathfrak{gl}_n(\mathbb{C})$ be a parabolic algebra of a parabolic group $P\subseteq GL_n(\mathbb{C})$. What is the difference among the nilradical of $\mathfrak{p}$, the ...
0
votes
1answer
46 views

Vector Space of Lie Algebra

Lie algebra $ \mathfrak{g} $ for a Lie group $ \mathcal{G}$ is closed under commutation. Also, the elements of Lie Algebra form a Linear Vector Space(LVS). Firstly, when is it allowed to define an ...
0
votes
1answer
19 views

Difference between the pairings $\text{Tr}(xy)$ and $\text{Tr}(x^t y)$

Let $\mathfrak{g}$ be the tangent space to $GL_n(\mathbb{C})$ at the identity. What is the difference between the two maps? Any subtle geometric or algebraic difference between the two pairings $$ ...
0
votes
0answers
31 views

Ideals in the unitary group

What would be examples of one-dimensional ideals in the lie algebra of the unitary group? Moreover, how would one show that it is in the tangent space of the center of the unitary group and that the ...
0
votes
1answer
21 views

Induced automorphism on a tangent bundle

I had a pretty simple question but was having trouble finding the answer anywhere. If I have an orthogonal matrix $A: \mathbb{R}^n \to \mathbb{R}^n$, it should induce an automorphism on the tangent ...
0
votes
1answer
25 views

Given tangent space of submanifold of Lie group, is it possible to recover the submanifold?

I have computed the tangent space of certain submanifolds (the unstable manifolds) of a Lie group at a particular point. I know that the exponential map lets us move between the Lie algebra and the ...
0
votes
1answer
15 views

Space of tangents of a matrix group G?

Given a smooth path A(t) through the identity in any matrix group G, how would one prove that the smooth path through any g in G, is of the form gA(t)? It is clear that gA(t) is differentiable and ...
2
votes
1answer
45 views

simple Lie groups

A Lie group is a group which is a smooth manifold such that the multiplication and inversion are smooth. When does a Lie group become simple? What is the difference between simple and semi-simple Lie ...
6
votes
1answer
68 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
0
votes
0answers
20 views

Bilinear form on the space of smooth complex valued functions.

Let $G$ be a Lie group and $h$ be the Hermitian bilinear form on smooth complex valued functions then how can we define bilinear form on the space of smooth complex valued functions.
3
votes
1answer
69 views

Baker–Campbell–Hausdorff formula for [exp(x),exp(y)]

Can someone provide a explicit (the first priority with leading orders, then the secondary consider as complete as possible, or) expansion like Baker–Campbell–Hausdorff formula for the commutator: ...
1
vote
1answer
25 views

Question on lifting the Weyl group into the group of inner automorphisms of $\mathfrak{g}$

I'm looking for some clarification of a statement that I found in Kac and Peterson's paper (112 realizations of the basic representation of the loop group of $E_8$). Let $\mathfrak{g}$ be a simple ...
2
votes
1answer
43 views

Computing Lie algebra of a subgroup

I will like to know how does one compute the Lie algebra of an abstractly given subgroup of a Lie group? Specifically, let $G = \mathrm{SO} ( n + 1, 1 )$ and consider the flow $$ g_t = ...
0
votes
0answers
59 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
1
vote
0answers
29 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

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
42 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
20 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
47 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 ...
0
votes
1answer
64 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 ...
2
votes
0answers
64 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
54 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 = ...
5
votes
3answers
72 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 ...
0
votes
1answer
36 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
80 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} ...
5
votes
2answers
96 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
35 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
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
vote
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
42 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 ...
1
vote
0answers
35 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 $U(n)\simeq SU(n)\times U(1)$?
1
vote
1answer
53 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 & ...
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 ...
0
votes
1answer
53 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?
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\\ ...