# Tagged Questions

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 ...

0answers
11 views

### Centralizers of connected linear group and its Lie algebra

If we have that $G$ is a connected linear group and $H<G$ with $\mathfrak{h}$ the lie algebra of $H$ and we define the centralizers of the elements in the following way: \$Z(H):=\{a\in G| ...
0answers
21 views

### Is the truncated exponential series for matrices injective?

If $k$ is a field of characteristic $p$, we can define a map $\exp:\mathfrak{gl}_n(k)\to GL_n(k)$ by: $$\exp(A)=\sum_{i=0}^{p-1}\frac{A^i}{i!}$$ In the answer to this question, we see that if ...
0answers
14 views

### Calculating the lie algebra of $SO(2,1)$

I am trying to calculate the Lie algebra of the group $SO(2,1)$ where this is defined as: $SO(2,1=\{X\in Mat_3(\mathbb{R})|X^t\eta X=\eta, \det(X)=1\}$ where $\eta$ is the matrix defined as: \left ...
1answer
46 views

### The dimension of $SU(n)$

$SU(n)$ denotes the special unitary group. I know its dimension should be $n^2-1$. However, I am trying to prove it and get a wrong result. I have no idea what is wrong with my proof. Therefore, I am ...
2answers
52 views

### Symplectic Form Preserved by Orthogonal Transformation

I'm trying to prove that the symplectic form $$\omega = d(\cos\theta) \wedge d\phi$$ is preserved by the action of $SO(3)$ on $S^2$ where $\phi$ and $\theta$ are spherical polars. Now $SO(3)$ simply ...
0answers
29 views

### Action of a Lie group on a coset of its subgroup

I am a physicist, so sorry for the lack of rigor. It is well known that a (say compact) Lie group $G$ acts naturally by left multiplication on the coset space $G/H$ where $H\subset G$ is its (Lie) ...
0answers
21 views

### Maximal compact subgroup of $GL_n(\mathbb C_p)$

It is known that the general linear group $GL_n(\mathbb Q_p)$ over the $p$-adic numbers has $GL_n(\mathbb Z_p)$ as a maximal compact subgroup and every other maximal compact subgroup of \$GL_n(\mathbb ...
1answer
20 views

### Proving that the Flag Variety $Fl(n;m_1,m_2)$ is connected.

I wish to prove that the flag variety $Fl(n;m_1,m_2) = \{ W_1 \subset W_2 \subset V | dimW_i = m_i \}$, for $0 \le m_1 \le m_2 \le n$ where V is an n-dimensional vector space over $\mathbb{C}$ and ...
0answers
10 views

### Is a quotient of maximal torus maximal for Lie groups?

I currently learning about Lie theory. Specifically, I am learning about maximal torus. However, I do not understand how these objects interact with quotients of subgroups. For instance if \$T\subseteq ...
1answer
32 views

### Truncated exponential map from $\mathfrak{gl}_n$ to $GL_n$

Let $k$ be a field of characteristic $p>0$. If $A$ is a nilpotent matrix in $\mathfrak{gl}_n(k)$, with $p>n$, then we can define the unipotent matrix: ...
0answers
18 views

### How to write down the maximal subgroups of $GL(9, \mathbb{C})$

I am wondering about the maximal subgroups of the group $GL(n^2, \mathbb{C})$. My motivation for wondering about these groups is a project (in its most general form) I am working on where I am trying ...
0answers
15 views

### are closed orbits of Lie group action embedded?

Consider a smooth action $G\curvearrowright M$ of a Lie group on a manifold. Suppose that an orbit $G\cdot p$ is closed. Is the orbit an embedded submanifold. In general we know that the orbits are ...
1answer
25 views

### Lie subalgebra, Lie subgroup and membership

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $H$ be a connected Lie subgroup with Lie algebra $\mathfrak{h}$. We have that $X \in \mathfrak{h}$ iff \$exp(tX) \in H \ \ \ \forall t ...
3answers
78 views

### Show that $\exp: \mathfrak{sl}(n,\mathbb R)\to \operatorname{SL}(n,\mathbb R)$ is not surjective

It is well known that for $n=2$, this holds. The polar decomposition provides the topology of $\operatorname{SL}(n,\mathbb R)$ as the product of symmetric matrices and orthogonal matrices, which can ...
0answers
58 views

### The Symplectic group is connected

Let $K = \mathbb{R}, \mathbb{C}$ be a field and consider the skew-symmetric matrix $$J = \left( \begin{matrix} 0 & I_n \\ -I_n & 0 \end{matrix} \right)$$ where $I_n$ is the unit matrix of ...
0answers
27 views

### Proof of Lie theorem on solvable Lie algebra

I am reading a book of Helgason. As you know, solvable Lie algebra $g \subset V= {\bf C}^n$ have a nonzero $v$ such that $v$ is an eigenvector of any element of $g$. I can follow the proof in ...
0answers
28 views

### To what extent are formulas obtained in one Lie group valid in another Lie group with an isomorphic Lie algebra?

In quantum optics, I am trying to explore the group generated by squeezing and rotation operators. These are closely related to area-preserving linear transforms, which they induce on the phase space, ...
1answer
73 views

### Show that an orthogonal group is a $\frac{n(n−1)]}2-$dim. $C^\infty$-Manifold and find its tangent space

The orthogonal group is defined as (with group structure inherited from $n\times n$ matrices) $$O(n) := \{X\in \mathbb{R}^{n\times n} : X^\text{t}X=I_n\}.$$ (i) Show that $O(n)$ is an ...
1answer
45 views

### Analogues of $SU(2)$ and $SO(3)$

The groups $SU_2(\mathbb{C})$ and $SO_3(\mathbb{R})$ are interesting in geometry, and there is a $2$-to-$1$ map from $SU_2(\mathbb{C})$ to $SO_3(\mathbb{R})$. There are finitely many finite groups in ...
0answers
5 views

### Is the application $D(R_p\circ \imath)(e):\mathfrak{h}\rightarrow T_p\mathcal{L}_p$ an isomorphism?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and $H\subseteq G$ a Lie subgroup with Lie subalgebra $\mathfrak{h}$. Consider the right translation $R_p:G\rightarrow G$ given by $R_p(g)=gp$. ...
1answer
30 views

### Does the equality $[u, v]=[X, Y](e)$ holds?

Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra and $\mathfrak{h}\subseteq \mathfrak{g}$ a vector subspace. I defined two smooth vector fields $X, Y:G\rightarrow TG$ setting $X(g)=DR_g(e)u$ and ...
0answers
25 views

### Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
0answers
27 views

### The simply-connectedness of quotient space

If $U$ is a Lie group with a closed subgroup $K$ such that both $U$ and $U/K$ are simply-connected, then can we conclude that $K$ is connected?
0answers
29 views

### Character of half-spin representation

Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum \sum x_1^{\pm ...
0answers
26 views

### Optimization of Möbius transformation

Say I have a family of points $(w_i, z_i)$ for $i=1,2,...,n$, and I wish to find $a,b,c,d$ such that $\sum_i \left|\frac{a z_i -b}{c z_i - d} - w_i \right|^2$ is minimized. I realize there are things ...
1answer
164 views

### Structure constants of Lie algebra

Let $(x^i)$ be a local coordinates system near identity of a Lie group $G$ such that $x(e)=0$. Suppose the multiplication has local form m(x_1,x_2)^k=x_1^k+x_2^k+\frac{1}{2}b_{ij}^k x^i_1 ...
1answer
38 views

### Complexification of the real lie algebra $\mathrm{sp}(m,n)$

I am unable to verify the fact that the complexification of the real lie algebra $\mathrm{sp}(m,n)$ is $\mathrm{sp}(2(m+n),\mathbf C)$, where $\mathrm{sp}(m,n)$ is the set of endomorphisms preserving ...
1answer
35 views

### Is this distribution involutive?

For two days I've been trying to show the following: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and consider the smooth distribution $$F=\{F_p=DR_p(e)\mathfrak{h}; p\in G\},$$ where ...
0answers
54 views

### Differentiation in group space

In a few physics papers (lattice gauge theory papers, to be more specific) I've seen the following definition for differentiation on group space  \frac{\partial}{\partial U} f(U) = ...
1answer
58 views

### Question about lie bracket..

Let $G$ be a Lie group with Lie algebras $\mathfrak{g}$ and let $\mathfrak{h}\subseteq \mathfrak{g}$ be a Lie subalgebra. Write $F_p=DR_p(e)\mathfrak{h}$, $p\in G$, where $R_p:G\rightarrow G$ given by ...
1answer
53 views

### Equality involving Lie Brackets

I have a question concerning Lie brackets: Consider the Lie bracket $$[, ]:\mathfrak{g}\times \mathfrak{g}\rightarrow \mathfrak{g},$$ where $\mathfrak{g}=T_eG$ is the Lie algebra of a Lie group $G$. ...
0answers
26 views

### The Lie algebra of the commutator subgroup

If $G$ is a connected Lie group with Lie algebra $g$, then is its commutator subgroup $[G,G]$ a closed subgroup with Lie algebra $[g,g]$?
1answer
49 views
+50

### Every principal $G$-bundle over a surface is trivial if $G$ is compact and simply connected: reference?

I'm looking for a reference for the following result: If $G$ is a compact and simply connected Lie group and $\Sigma$ is a compact orientable surface, then every principal $G$-bundle over $\Sigma$ ...
0answers
37 views

### Why these two groups are closed in two other?

I have no strategy to show that following groups are closed in after group. $$K=\{g=(g_{i,j}\in U(n+1))\mid g_{2,1}=\ldots g_{n+1,1}\}\quad in \quad U(n+1)$$ U(n+1)\quad in \quad\{A = (a_{ij}) \in ...
1answer
24 views

### What is the number of non-compact generators of $\operatorname{so}(p, q)$ and $\operatorname{su}(p, q)$?

Setting $n = p + q$, the total number of generators of $\operatorname{so}(p, q)$ or $\operatorname{su}(p, q)$ is respectively $n(n - 1) /2$ and $n^2 - 1$. But what is the number of non-compact ...
1answer
27 views

### The closed subgroup of Lie group

$G$ is a connected Lie group with Lie algebra $g$ and $l$ is an abelian ideal of $g$. If $K$ is the connected Lie subgroup of $G$ with the Lie algebra $l$, then is $K$ necessarily closed in $G$?
1answer
55 views

### Tangent space at the identity element of a lie group

Let G be a lie group . we know a Lie group is a group with a smooth manifold structure s.t both the multiplication map $m$ and group inversion map $i$ are smooth . Now by identifying ...
1answer
37 views

### Isomorphisms of the Lorentz group and algebra

I'm trying to read a few books on QFT and some seem to say the Lorentz algebra obeys $\mathfrak{so}(1,3)\otimes \mathbb{C} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ while others say ...
0answers
16 views

### concerning coadjoint representation

Let $\xi$ be the vector field on $\frak{g}^*$ (dual of Lie algebra) which correspond to element $X$ of the Lie algebra $\frak{g}$. Then why have we $\xi(F)=K_*(X)F$ where here $K=Ad^*(g)$ is ...
1answer
39 views

### Projective linear group - solvable

Let $q\geq 5$ and let PGL(2,q) be the projective general linear group. Question Do there exists a $q$ such that PGL(2,q) is solvable?
0answers
26 views

### Orbits of the action of G/H

Let $G \subset Iso(M)$ be a Lie group which acts on a (semiriemannian) manifold $M$ properly and smoothly. Let we know the orbits of the action. Suppose that $H$ is a discrete central subgroup of $G$ ...
0answers
25 views

### Classifying all rank 2 and 3 root systems

I am working with the representation theory of complex simple Lie algebras, and have a question: It is intuitively clear that the root systems $A_1\times A_1$, $A_2$, $B_2$, and $G_2$ comprise all ...
0answers
29 views

### a question about G-Manifolds

I am looking for a clear reason for following fact: Why a $G$-invariant differential form $\omega$ on a homogeneous $G$-manifold $M=G/H$ is uniquely determined by its value at the initial point $m_0$ ...
0answers
16 views

### Why is the dual space of Cartan subalgebra an irreducible representation of Weyl group

it is proposition 14.31 in Fulton-Harris book. The proof goes like this. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and assume $\mathfrak{z}\subseteq\mathfrak{h}^*$ were preserved ...
0answers
30 views

### Usage and determination of “rank” and “dimension” of groups & representations

Physicist here. I seem to see conflicting statements about the rank of some groups I've come across lately. A paper I'm reading states that $SO(6)$ is rank 3 and therefore its Cartan subalgebra ...
0answers
32 views

### Finding the dimension of the symplectic group

How do you find the dimension of the symplectic group $Sp(2n,\mathbb{R})$? $Sp(2n,\mathbb{R})\subset Gl(2n,\mathbb{R})$ is the group of invertible matrices $A$ such that $\omega = A^T\omega A$, where ...
1answer
83 views

### Proof that $U(n)$ is connected

I'm trying to prove that $U(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I\}$ is connected, but most of the proof comes down to proving that $SU(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I$ and \$ ...
0answers
70 views

### How many discrete subgroups does the Heisenberg group have?

Is there an easy way to describe an arbitrary discrete group in the Heisenberg group? I figured that at least the family  \begin{pmatrix} 1 & x\mathbb Z & z\mathbb Z\\0&1&y\mathbb ...
1answer
19 views

### Is this a set of generators for the conformal group of Minkowski space?

My physics textbook asserts that the group of maps $f: M \rightarrow M$ ($M$ is the Minkowski space, i. e. $\Bbb R^4$ with the pseudonorm $||x||=x_0^2-x_1^2-x_2^2-x_3^2$ and scalar product \$x\dot{} ...
1answer
30 views

### Elements of order 2 in a Weyl group.

I would like to prove that any element of order 2 in a Weyl group is the product of commuting root reflections. I am told that the proof should be by induction on the dimension of the -1 eigenspace. ...
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