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 ...
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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| ...
1
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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 ...
1
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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: $$\left ...
3
votes
1answer
46 views
The dimension of $SU(n)$
$SU(n)$ denotes the special unitary group. I know its dimension should be $n^2-1$. However, I am trying to prove it and get a wrong result. I have no idea what is wrong with my proof. Therefore, I am ...
2
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2answers
52 views
Symplectic Form Preserved by Orthogonal Transformation
I'm trying to prove that the symplectic form
$$\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 ...
1
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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) ...
2
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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 ...
4
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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 ...
0
votes
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 ...
1
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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:
...
1
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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 ...
1
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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 ...
1
vote
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 ...
3
votes
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 ...
3
votes
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) $$
where $I_n$ is the unit matrix of ...
1
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0answers
27 views
Proof of Lie theorem on solvable Lie algebra
I am reading a book of Helgason.
As you know, solvable Lie algebra $g \subset V= {\bf C}^n$ have a nonzero $v$ such that
$v$ is an eigenvector of any element of $g$.
I can follow the proof in ...
2
votes
0answers
28 views
To what extent are formulas obtained in one Lie group valid in another Lie group with an isomorphic Lie algebra?
In quantum optics, I am trying to explore the group generated by squeezing and rotation operators. These are closely related to area-preserving linear transforms, which they induce on the phase space, ...
0
votes
1answer
73 views
Show that an orthogonal group is a $\frac{n(n−1)]}2-$dim. $C^\infty$-Manifold and find its tangent space
The orthogonal group is defined as (with group structure inherited from $n\times n$ matrices)
$$O(n) := \{X\in \mathbb{R}^{n\times n} : X^\text{t}X=I_n\}.$$
(i) Show that $O(n)$ is an ...
4
votes
1answer
45 views
Analogues of $SU(2)$ and $SO(3)$
The groups $SU_2(\mathbb{C})$ and $SO_3(\mathbb{R})$ are interesting in geometry, and there is a $2$-to-$1$ map from $SU_2(\mathbb{C})$ to $SO_3(\mathbb{R})$. There are finitely many finite groups in ...
0
votes
0answers
5 views
Is the application $D(R_p\circ \imath)(e):\mathfrak{h}\rightarrow T_p\mathcal{L}_p$ an isomorphism?
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and $H\subseteq G$ a Lie subgroup with Lie subalgebra $\mathfrak{h}$. Consider the right translation $R_p:G\rightarrow G$ given by $R_p(g)=gp$. ...
2
votes
1answer
30 views
Does the equality $[u, v]=[X, Y](e)$ holds?
Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra and $\mathfrak{h}\subseteq \mathfrak{g}$ a vector subspace. I defined two smooth vector fields $X, Y:G\rightarrow TG$ setting $X(g)=DR_g(e)u$ and ...
4
votes
0answers
25 views
Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$
If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
3
votes
0answers
27 views
The simply-connectedness of quotient space
If $U$ is a Lie group with a closed subgroup $K$ such that both $U$ and $U/K$ are simply-connected, then can we conclude that $K$ is connected?
2
votes
0answers
29 views
Character of half-spin representation
Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum
$$\sum x_1^{\pm ...
3
votes
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 ...
6
votes
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
$$m(x_1,x_2)^k=x_1^k+x_2^k+\frac{1}{2}b_{ij}^k x^i_1 ...
2
votes
1answer
38 views
Complexification of the real lie algebra $\mathrm{sp}(m,n)$
I am unable to verify the fact that the complexification of the real lie algebra $\mathrm{sp}(m,n)$ is $\mathrm{sp}(2(m+n),\mathbf C)$, where $\mathrm{sp}(m,n)$ is the set of endomorphisms preserving ...
1
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1answer
35 views
Is this distribution involutive?
For two days I've been trying to show the following: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and consider the smooth distribution $$F=\{F_p=DR_p(e)\mathfrak{h}; p\in G\},$$ where ...
1
vote
0answers
54 views
Differentiation in group space
In a few physics papers (lattice gauge theory papers, to be more specific) I've seen the following definition for differentiation on group space
$$
\frac{\partial}{\partial U} f(U) = ...
1
vote
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 ...
1
vote
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 $G$. ...
1
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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 $[g,g]$?
3
votes
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 $\Sigma$ ...
0
votes
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 ...
0
votes
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 ...
0
votes
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$?
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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?
1
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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$ ...
0
votes
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 ...
-1
votes
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$ ...
1
vote
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 ...
1
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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 ...
2
votes
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 ...
3
votes
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 $ ...
3
votes
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 ...
1
vote
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{} ...
0
votes
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.
...
