0
votes
1answer
21 views

Lie algebra for SO(3) as a skew symmetric matrix

How can I show that the associated lie algebra for SO(3) is the set of all 3 dimensional skew-symmetric matrices?
0
votes
0answers
14 views

Convergence of Baker-Cambpbell-Hausdorf for compact groups

It is well known that the Baker-Campbell-Hausdorf formula doesn't need to converge for general elements of a Lie algebra, resp. for matrices with norms larger then 1. On the other side, if $G$ is a ...
0
votes
0answers
32 views

Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ ...
3
votes
1answer
53 views

Commuting quaternions

I tried to solve the following exercise, please could somebody tell me if I did it right?: Prove that non-real elements $x,y \in \mathbb H$ commute if and only if their imaginary parts are ...
1
vote
0answers
36 views

Matrix multiplication in quaternions is not necessarily linear

I tried to show by example that matrix multiplication for quaternionic matrices is is not necessarily $\mathbb H$-linear. If $A \in M_n(\mathbb H)$ is a quaternionic matrix and $x$ is a vector in ...
3
votes
3answers
74 views

Why not $SL_n (\mathbb R)$ in this exercise

I just solved the following exercise: Let $SL_2(\mathbb Z)$ denote the set of $2\times2$ matrices with integer entries and determinant $1$. Prove that $SL_2(\mathbb Z)$ is a subgroup of ...
1
vote
0answers
27 views

$SO(n)$ is connected, alternative form

I have the following exercise: Show that $SO(n)$ is connected, using the following outline: For the case $n = 1$, there is nothing to show, since a $1\times 1$ matrix with determinant one must be ...
2
votes
0answers
22 views

Linearly independent skew symmetric complex matrices having the least eigenvalues

Question: Let $A$, $B$ be two $5 \times 5$ (or $7 \times 7$) skew-symmetric complex matrices (i.e. $A^t = -A$), and suppose that $$ \forall t,s \in \mathbb{C}, \quad M(t,s):=(tA+sB)^*(tA+sB) \text{ ...
3
votes
1answer
58 views

tangent/normal space to set of symmetric isospectral matrices

Let $\Lambda = \{\lambda_1, \ldots, \lambda_n\}$ be a set of $n$ distinct real numbers. $M_n(\mathbb{R})$ denotes the set of all $n \times n$ real matrices, and for $B\in M_n(\mathbb{R})$, $B^T$ ...
0
votes
0answers
13 views

Reference request: Tensor Product of $m$ $SU(N)$ algebras

I'm working on quantum mechanics of linear $SU(N)$ chain of sites. Specifically, I would like to study a Tensor product of $m$ Lie algebras $\mathfrak{s}\mathfrak{u}(N)$ $$ V\in SU(N)\\ W = ...
0
votes
0answers
15 views

Dual basis to $e_{i+1}-e_{i} \in \ker ((1,1,…1)^\vee\in(\Bbb E^{n+1})^\vee)$

Studying the root system $A_n$ given by the simple roots $v_i:=e_{i+1}-e_i \in \Bbb E^{n+1}/\Bbb R(1,1,...,1)$ for $i = 1,...,n$, I came across the following dual basis: $v_i^\vee:= ...
0
votes
1answer
25 views

SO(n) is parallelizable

Prove that $SO(n)$ is parallelizable. How would I go about showing this? My supervisor could not help me with this problem, and I am stumped.
0
votes
1answer
66 views

Centralizer of $SO(n)$

Given the set $M(n,\mathbb C)$ of all complex $n\times n$ matrices, what's the centralizer of $SO(n)$ in $M(n,\mathbb C)$? For $n=2$, the centralizer must be the matrices $A$ such that $RA=AR$ where ...
3
votes
0answers
66 views

Is there a name for the group of complex matrices with unimodular determinant?

Does the group $$ G = \left\{ A \in \mathbb{C}^{n \times n} : |\det(A)| = 1 \right\} $$ have a name? It obviously contains the unitary group $U(n)$ and the special linear group $SL(n,\mathbb C)$. ...
4
votes
2answers
127 views

Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?

This might be a dumb question; I know only enough group theory to be able to ask dumb questions. Ken W. Smith has pointed out that one way to get intuition about the determinant is to observe that it ...
0
votes
1answer
69 views

homework: proof regarding skew-symmetric nondegenerate bilinear form

I have this proof for a homework exercise, which I can't figure out how to solve.. Let $V$ be a vector space over $F$ and let $B$ be a skew-symmetric or symmetric nondegenerate bilinear form on $V$. ...
1
vote
1answer
31 views

determinating the signature of a bilinear form

As as part of my course in Lie groups, I need some help to determine the signature of the form $B(x,y) = $$ \sum_{i=1}^{n}\ x_iy_{n+1-i}$ More than anything, I would like to understand how to ...
0
votes
1answer
55 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 ...
2
votes
2answers
60 views

Dimension of $SO_n(\mathbb{R})$

Is there a simple proof that the dimension of $SO_n(\mathbb{R})$, a.k.a the group of rotations in $n$-dimensional space is $(n-1)n/2$? It would be great to see some proofs based only on the ...
0
votes
0answers
22 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.
0
votes
0answers
64 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
2
votes
0answers
29 views

Existence of maximal tori in infinite-dimensional Lie groups

Reading about Lie groups and maximal tori I came up with a lemma that states that any Lie group $G$ has maximal tori. The proof goes like this: firstly, it is proven that if $H \subset G$ is a proper ...
2
votes
1answer
25 views

Questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$.

I have some questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$. Since $\mathbb{F}_q$ is not algebraically closed, it is possible that some $g \in GL_2(\mathbb{F}_q)$ has eigenvalues which ...
1
vote
0answers
54 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
1
vote
2answers
75 views

Quaternion representation of rotations

How would one show that 1/3 turns correspond to the eight antipodal pairs among the 16 quaternions : $$ \pm \frac{1}{2} \pm \frac{i}{2} \pm \frac{j}{2} \pm \frac{k}{2} $$ knowing that the rotation ...
0
votes
0answers
98 views

Why Bruhat decomposition in $GL_n$ case is the Gauss decomposition?

Gauss decomposition of a matrix is also called LU decomposition. Let $A$ be a matrix. Then $A=LU$ for some lower triangular matrix $L$ and upper triangular matrix $U$. This can be obtained using Gauss ...
11
votes
2answers
164 views

Is $SO_n({\mathbb R})$ a divisible group?

The title says it all ... Formally, if $SO_n(\mathbb R)=\lbrace A\in M_n({\mathbb R}) |AA^{T}=I_n, {\sf det}(A)=1 \rbrace$ and $W\in SO_n(\mathbb R)$, is it true that for every integer $p$, there is ...
0
votes
1answer
30 views

Matrix multiplication in $SO(3)$ that fixes row

I want to find all matrices $G \in SO(3)$ that do not change the first row of elements in $SO(3)$ when right multiplying by $G$, i.e. $$ \{ G \in SO(3): \forall A \in SO(3) \quad A = \begin{pmatrix} ...
3
votes
3answers
180 views

What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
0
votes
1answer
51 views

What is this dimension?

What is the real dimension of the cone of $2$ by $2$ Hermitain matrices with at lease one eigenvalue that is $0$?
2
votes
2answers
78 views

How to show that exp is a diffeomorphism between symmetric reals and positiv definite matrices?

I am looking for an easy proof of the fact that the exponential function is a diffeomorphism between the finite dimensional vector space of symmetric real nxn-matrices and the open subset of positive ...
1
vote
1answer
102 views

A general element of U(2)

The group U(2) is the group of all $2\times 2$ matrices such that $U^\dagger U=I$. Evidently it has $4$ real parameters, and can be represented as: $$ U(2) = \{\begin{bmatrix}a&b\\ 0&d ...
4
votes
1answer
95 views

A Neat Rotation Matrix Identity

Let $\mathbf{R}_i$ be $N$ rotation matrices that represent a rotation around axes $\mathbf{\omega}_i$ by an angle $|\mathbf{\omega}_i|$. Now say we know that the product of these matrices is unity, ...
0
votes
1answer
45 views

Structural Question about Elements of $SO(2n)$.

Let $R_\theta$ denote the appropriate element of $SO(2)$, as is conventional. Let $n$ be a positive integer, and consider $D = \text{diag}(R_{\theta_1}, \dots, R_{\theta_n}) \in SO(2n)$. For ...
0
votes
1answer
27 views

Cofactors and conjugates of $SU(3)$.

I was playing around with some equations and noticed the following: Let $A$ be an element of $SU(3)$ with components $A_{ij}$. If $C_{ij}$ is the $(i,j)$ cofactor of $A$ then $C_{ij} = ...
3
votes
1answer
58 views

Is it true that stabilizer in $O(n)$ of a rank $k$ matrix is isomorphic to $O(n-k)$?

Let $X\in M_{n, k}(\mathbb R)$ such that $\textrm{rank}(X)=k$ and,$$O(n)_X:=\{A\in O(n): AX=X\}.$$ Notice $O(n)_X$ is a subgroup of $O(n)$. Is it true that $O(n-k)\cong O(n)_X$? Here $O(n)=\{A\in ...
3
votes
1answer
123 views

Find the tangent space of $\mathrm{Aff}(n)$

Find the tangent space of $\mathrm{Aff}(n)$. see Proof: Tangent space of the general linear group is the set of all squared matrices $\mathrm{Aff}(n)$ is the set of all matrices of the form $$ ...
0
votes
1answer
42 views

Differences of skew symmetric matrices

Let $A$ be an invertible real skew-symmetric matrix, and consider the difference $A_R:=RAR^{-1}-A$, for orthogonal $R$. Is it true that $A_R$ is either zero or invertible? Does the answer depend on ...
1
vote
1answer
41 views

What 1D $\mathbb{C}$-Subspaces are Stabilized by Elements of a Specific 2-Torus in $SO(7)$?

Consider the 2-torus $T \subset SO(7)$ defined by $T = \left\{ \mathrm{diag}(R_{\theta_1}, R_{\theta_2}, R_{-(\theta_1 + \theta_2)}, 1) \mid \theta_1, \theta_2 \in \mathbb{R} \right\}$, where ...
3
votes
1answer
74 views

How to determine all conjugacy classes in the complex orthogonal group $O(n,\mathbb{C})$ of finite order $k$

I was wondering how one would go about determining the conjugacy classes of the complex orthogonal group $O(n,\mathbb{C})$ of some finite order $k$. That is, if $[A]$ is the conjugacy class of $A\in ...
3
votes
2answers
66 views

action of $O(n,\mathbb{R})$ on ${S}^{n-1}$

Is the action of $O(n,\mathbb{R})$ on ${S}^{n-1}$ transitive? I think this is true as orthogonal matrices are supposed to rotate and keep the length fixed, but how do I prove this? EDIT: Based on ...
2
votes
1answer
79 views

Proofs that: $\text{Sp}(2n,\mathbb{C})$ is Lie Group and $\text{sp}(2n,\mathbb{C})$ is Lie Algebra

Consider following Lie Group: $$ \text{Sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}(\mathbb{C})\mid J=g^TJg\}\quad\ where\quad J=\begin{pmatrix} 0 & 1_n \\ -1_n & 0 \end{pmatrix} $$ And the ...
0
votes
1answer
49 views

Range of the exponential map

I need to build a matrix $A\in Gl^+_n(\mathbb{R})$ ($det(A)>0$) that is not an exponent, i.e. there is no $B\in Mat_n(\mathbb{R})$ such that $A=exp(B)$. Could you give me a hint to note some ...
3
votes
1answer
64 views

How to show $SL_{n}(\mathbb{R})=\bigsqcup_{w\in W}LwU$ where L (or U) are lower(or upper) triangular matrix?

I'd like to ask a homework problem that causes me many troubles for days. The problem is like below : Let W denote the subgroup of permutation matrices in $SL_{n}(\mathbb{R})$. Show the following ...
3
votes
3answers
115 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 ...
1
vote
1answer
44 views

Why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?

Let $G=SO(n,R)$ be a Lie group and $\mathbb{g}$ its lie algebra. Take $X\in \mathbb{g}$. Then why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?
3
votes
1answer
113 views

Finding the Lie Algebra of a Lie Group

I am having a hard time finding the set of all $X \in M(n, \mathbb{R})$ such that $e^{tX^T}Be^{tX} = B$ for all $t \in \mathbb{R}$. where $b$ is any matrix in $M(n, \mathbb{R})$.
4
votes
1answer
108 views

orthogonal group of a quadratic vector space

I am reading about the orthogonal group $O(V)$ of a real finite dimensional quadratic vector space $(V,Q)$ with $Q$ nondegenerate. By definition $$O(V)=\{f:V\mapsto V |\quad Q(f(v))=Q(v) \quad \forall ...
7
votes
2answers
651 views

Two Definitions of the Special Orthogonal Lie Algebra

I am encountering two definitions of the special orthogonal lie algebra, and I would like to know if they are equivalent, and if there are advantages to working with one over the other. If we begin ...
1
vote
1answer
178 views

Determining which maps are isomorphisms

1) Let $G=G'=\{(a,b)\mid a,b \in \mathbb{R}, a, b \ne 0\}$ with group operation $(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2)$. Let $\phi (a,b) = (b^{-1}, ab^2)$. My solution: 1-1: Suppose ...