A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group. Consider using with the (group-...

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6
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0answers
136 views

Generators of so(7)

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is isomorphic to $\mathfrak{so}(7)$. Can ...
1
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1answer
25 views

What is the root system and the Weyl group of the group spin$(2n)$?

Reading on root systems and Weyl groups, unfortunately I am highly confused when it comes to the spin-groups (the two-fold universal cover of SO$(2n, \mathbb{C})$, realizable as a quotiënt in a ...
2
votes
1answer
35 views

Reductive homogeneous spaces

If $G$ is a connected Lie group and $K$ is a closed subgroup of $G$ then $G/K$ is a homogeneous space. If $\frak g,k$ are the lie subalgebras of $G,K$ resp. Then under the projection $\pi:G\rightarrow ...
0
votes
1answer
62 views

Cartan decomposition diffeomorphism at the level of (compact) groups

I am trying to understand the Cartan decomposition in the case of a compact group $G$ with subgroup $K$, such that their respective Lie algebras $(\mathfrak{g},\mathfrak{k})$ correspond to a Cartan ...
0
votes
0answers
45 views

Is it possible to study Lie algebras without knowing too much of representation theory?

There's a course on Lie Groups that I'd like to take, but it seems that for various reasons it's a good idea to take Lie algebras along with it. But after having a brief look at the contents of the ...
1
vote
1answer
35 views

A noncompact Lie group need not have any nontrivial tori

I don't understand why the following statement is true. 'A noncompact Lie group need not have any nontrivial tori (e.g. $\mathbb{R}^n$). Taking $n=2$, we get $\mathbb{R}^2$. Now consider the set of ...
0
votes
0answers
8 views

Irreducible tensor for fundamental representation of SU(N=3)

I am trying to calculate the singlets of the tensor product $N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^*$. I know that $N_c \otimes N_c^*=1\oplus (...
2
votes
0answers
32 views

Introduction to flag manifolds

What is a good self-contained introduction to the geometry of complex flag manifolds?
0
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1answer
42 views

Riemannian connection on Lie groups

Let $G$ be a Lie group with a bi-invariant metric. Then, the Riemannian connection is given by $\nabla_XY=\frac1 2 [X,Y]$ for all $X,Y\in \mathfrak g$. In the proof: Since $\langle X,Y\rangle$ is ...
1
vote
1answer
39 views

Using isometric group to describe E7.

I read John C. Baez's paper, The Octonions, and I am wondering the following statement: $$E_7\simeq Isom(\mathbb{(H\otimes O)P}^2).$$ In his contents, I can only figure out $$E_7\hookrightarrow Isom(\...
1
vote
0answers
29 views

Partial differential equation transformation

Consider the partial differential equation $$i \frac{\partial \psi}{\partial t} - \left(i\nabla + \mathbf{A} \right)^2 \psi = 0 \tag{1}$$ for the scalar function $\psi(x,y,z)$ and the vector ...
3
votes
1answer
37 views

The group $\mathrm{Diff}(F)$ and transition functions of a fibre bundle.

Let $M$ and $F$ be differentiable manifolds, and let $F\to E\to M$ be a differentiable fibre bundle over $M$. A trivialising cover $\{(U_i,\phi_i)\,|\,i\in I\}$ of $M$ determines a set $\{t_{ij}:U_{ij}...
1
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0answers
15 views

Equation for plugging in right-invariant vector fields in canonical connection?

Consider a matrix Lie group $G$ with Lie algebra $\frak g$ identified with left-invariant vector fields $\mathcal L(G)$. The $0$-connection is given by: $$ \nabla_{X^l}{Y^l}=\frac{1}{2}[X^l,Y^l]=\frac{...
0
votes
1answer
29 views

Why the induced metric from the lie algebra of lie group $G$ is left invariant.

we say that a Riemannian metric on $G$ is left invariant if $<u,v>_y = <d(L_x)_y u,d(L_x)_y v>_{L_x(y)}$ to introduce a metric on $G$, take any arbitrary inner product $< , >_e$ on ...
0
votes
0answers
28 views

Limit of the commutator of two elements?

Given a Lie group $G$ such the $\mathfrak{g}$ denoted its Lie algebra. Let $[g,g']_{G}$ the commutator of two elements $g,g' \in G$ and denoted by $[X,X']_{\mathfrak{g}}$ the Lie bracket of two ...
1
vote
0answers
25 views

compact lie group -> real analytic orbits in $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Now, $G$ also acts on $\mathfrak{g}^*$, the dual of the Lie algebra, by the coadjoint-action. My question now is: are ...
3
votes
1answer
52 views

Symmetric decompositions of $SU(2)$ representations.

Let us consider the representation theory of $SU(2)$. There is a unique irreducible representation of dimension $n$ for each $n \ge 1$, which we will denote $\mathbf{n}$, with the defining $2$-...
1
vote
1answer
61 views

Irreducible representations of Heisenberg group

Lately, I've been struggling with the following problem. Let $H$ be the 3 dimensional Heisenberg group and let $\rho:H\to\text{GL}(n,\mathbb{C})$ be a irreducible representation. Show that $n=1$. I ...
1
vote
1answer
24 views

what if the infinitesimal generator of a vectorfield vanishes?

Let $(M,g)$ be a riemannian manifold and $H$ a Lie group acting on $M$. Denote by $l \colon H \times M \to M$ and $l_h \colon M \to M$ the action of $H$ on $M$. Now $H$ acts on $TM$ by derivations, ...
0
votes
0answers
13 views

About exceptional Lie group E6

How to show the group of determinant preserving linear transformations of $ z $ is isomorphic to $$ \{a \in Isom_\mathbb{C}(z^\mathbb{C},z^\mathbb{C})|det(aX)=det(X),<aX,aY>=<X,Y>\} $$ ...
-2
votes
1answer
74 views

Is the alternating group a lie group [duplicate]

Is the alternating group a lie group. If so what is the lie algebra corresponding to it? This is not a homework questions. I need the dimension of the lie algebra (if one exists) to prove some ...
1
vote
0answers
17 views

Finite dimensional irreducible representations of Sp(2).

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...
2
votes
1answer
47 views

If G is a compact semisimple Lie group, then its Killing form is negative definite

Theorem: If $G$ is a compact semisimple Lie group, then its Killing form is negative definite. In its proof: Since $G$ is compact there is an Ad-invariant inner product on $\mathfrak g$. Since each ...
2
votes
0answers
61 views

Involutions and Representation of Lie Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$. Is easy to see that ...
1
vote
0answers
26 views

A quotient by a discrete normal subgroup is locally isomorphic to the group itself

Let $G$ be a connected topological group and let $\Gamma$ be a discrete normal subgroup of $G$. Then why $G$ and $G/\Gamma$ are locally isomorphic?
0
votes
0answers
28 views

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$?

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$? This question is inspired by noting that if we have a Hamiltonian Lie group action $G\curvearrowright (M,\...
3
votes
1answer
61 views

Cartan decomposition of SO(2n)

I am trying to understand the Cartan decomposition theory, on the following example : $G=SO(2n)$, and $K=U(n)$, and I'm interested in the manifold $G/K$ (an hermitian symmetric space). 1) How do we ...
1
vote
0answers
28 views

By which the Heisenberg group is introduced? [closed]

I want to know who was the first who introduced the Heisenberg group and in what year. In the Wikipedia there is just an indication that this group was named in honor of the famous German physicist ...
3
votes
0answers
54 views

Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup?

Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts? https://en.wikipedia.org/wiki/Lie_group#...
1
vote
2answers
42 views

The Lie Algebra of $O(n)$ is the set of $n \times n$ skew-symmetric matrices

I'm trying to show that the Lie Algebra for $O(n)$ is the set of $n \times n$ skew-symmetric matrices. Here is what I have so far. Since $O(n)$ is the union of two disjoint subsets, the matrices with ...
0
votes
0answers
10 views

Graph properties of Bruhat order for the general linear Lie algebra $\mathfrak{gl}$ on $\mathbb{Z}^n$

Let $P = \oplus_{i\in \mathbb{Z}}\mathbb{Z}\epsilon_i$ the free abelian group of infinite rank. Then we have a natural partial order $\leq'$ on $P$, that is, $a \leq' b $ if and only if $b \in a+\sum_{...
0
votes
1answer
44 views

Reference request : manifolds and transitive lie group actions

I would like to show that any manifold $M$ with a transitive ation from a Lie group $G$ is diffeomorphic to $G/H$ where $H$ is the stabilizer of an element in $M$. Do you know any reference where I ...
1
vote
1answer
49 views

Representation of of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$

Certain part in my textbook implies that a representation of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$, where $S \in \mathbb Z$, is possible. I am trying to find a path that leads to this ...
1
vote
0answers
32 views

The linear Lie algebra of a closed linear group is closed

I was reading Knapp's Lie groups beyond an introduction and, in the first pages, he shows that the set of all tangent vectors to a given closed linear group $G$ at the identity matrix, that is $$\...
2
votes
1answer
19 views

Is it true that if all $G^\circ$ - orbits are closed in $X$ then all $G$ - orbits are closed in $X$?

Let $G$ be a Lie group acting continuously on a topological space $X$. Let $G^\circ$ be the connected component of the identity element of $G$ and let $[G:G^\circ]$ be finite. Then is the following ...
4
votes
1answer
214 views

restriction of $SL(2,R)$ representation to $SO(2)$

Let $d \geq 1$ and let $V$ be a $2d+1$ irreducible representation of $SL(2,\mathbb{R})$. We know the irreducible representations of $SO(2)$ are the $2$ dimensional spaces $V_k$ with the map $$\rho_k : ...
1
vote
1answer
31 views

The isotropy of the action of $SU(3)$ on $\mathbb CP^2$

Consider the action of $SU(3)$ on the complex projective plane $\mathbb CP^2$. How we can find the isotropy group?
4
votes
1answer
91 views

$E_8$ and theta functions

The root lattice $\Gamma_8$ of the exceptional Lie algebra $E_8$ is an eight-dimensional lattice which consists of lattice points in $\mathbb{R}^8$ which with respect to an orthonormal basis $e_1, \...
2
votes
1answer
58 views

The isotropy of the complex projective plane for the action of $SU(3)$

If we consider the action of the compact real form $SU(3)$ of $SL(3,\mathbb C)$ on the space $\mathbb C^3$. Since the action is transitive, how to find the stabilizer $G_x$? Is it useful to find ...
0
votes
3answers
53 views

If a Lie Algebra is solvable, is the corresponding Lie group solvable in the group theoretic sense?

I just started working with Lie Algebras with a professor. The way we defined them is probably the standard way; treat Lie Algebras as tangent spaces at the identity of the Lie Group. Now, consider ...
3
votes
0answers
33 views

Classification of irreducible (g,K)-modules for other g than sl2

Harish Chandra showed how to associate to an admissible representation $(\pi,V)$ of a real semisimple Lie group $G$ the so-called Harish-Chandra module $V_K$ of $K$-finite vectors in $V$. This is a $(\...
0
votes
1answer
34 views

A simple question for lie group

I know the following is wrong. But I don't see where is wrong. Let's begin by proving $\frac{d}{dt}(A(t)\cdot B(t))=(\frac{d}{dt}A(t))\cdot B(t)+A(t)\cdot\frac{d}{dt}B(t)$, where $A(t)$, $B(t)$ are ...
0
votes
0answers
22 views

real analytic structure on liegroup

Let $G$ be a Liegroup of $C^2$. I read, that if $G$ is compact, there exists a unique structure, such that $G$ is a real analytic manifold and the canonical action (left or right) is real analytic. ...
0
votes
1answer
40 views

Left and right invariant vector fields

I'm following Woodhouse's book on geometric quantization and I'm stuck with this problem. Let $R_A$ and $L_A$ be right and left invariant vector fields such that $R_A(e)=A=L_A(e)$, where $A$ is an ...
1
vote
1answer
30 views

How do we give G/T a symplectic structure

I am new in those staff and I can't find anything introductory explaining those stuff. I am interested in the following. Given $G$ a compact Lie group and consider it's maximal torus $T$. Then I have ...
0
votes
1answer
27 views

Why is the quotient map $G\rightarrow G/T$ a fibration?

I have just learnt about fibrations and I saw somewhere the following. Given a compact lie group $G$, one can consider a maximal torus $T$ of $G$ then the quotient map $G\rightarrow G/T$ is a ...
0
votes
1answer
26 views

Does equivariant property commute with matrix exponential?

Given a vector space $V$ and endomorphism $M : V \to V$ we can define the exponential of $M$: $\exp(M) = \sum_{k=0}^{\infty} \frac{1}{k!}M^k$. This has the property that it commutes with conjugation ...
0
votes
0answers
17 views

Laplacian in matrix form?

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
3
votes
2answers
34 views

Möbius Band Bundle $(Mo,\mathbb{S}^1,\text{proj}_1,\mathbb{R}) $ is not a Principal $\mathbb{R}$-bundle

This is claimed in various places. The problem seems to be with finding a free and transitive group action that has the fibers of $Mo$ as its orbits. I construct $Mo$ as $$ Mo = \mathbb{S}^1 \times \...
0
votes
2answers
53 views

For compact Lie groups, is $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$ equivalent to being semisimple?

In Ana Cannas da Silva’s Lectures on Symplectic Geomertry, page 167, semisimplicity is defined in the restricted setting of compact Lie groups by A compact Lie group $G$ is semisimple if $[\...