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{ ...
2
votes
2answers
66 views

Naive question about the group $SU(n)$?

As usual, let $SU(n)$ represent the set of all the $n\times n$ unitary matrices with determinant $1$. It's easy to show that any matrix $U$ takes the form $U=e^{iA}$ ($A$ is a $n\times n$ traceless ...
2
votes
1answer
36 views

Skew-Symmetric after base change symmetric?

Are there invertible matrices $A,B \in \textrm{GL}(\mathbb{C}^3)$ such that for every skew-symmetric matrix $S \in \textrm{Mat}_{3 \times 3} (\mathbb{C})$ the matrix $A \cdot S \cdot B$ is symmetric? ...
0
votes
0answers
18 views

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth.

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth. So where I am starting is by extending $Ad : G \rightarrow ...
0
votes
0answers
18 views

Double cover $Sp(1) \times SO(3) \rightarrow SO(4)$

I am working on understanding double covers at the moment. And I have come across a few double covers such as $Sp(1) \times Sp(1) \rightarrow SO(4)$ and $Sp(1) \rightarrow SO(3)$. And it seems to me ...
5
votes
2answers
130 views

traceless matrices

The fact that $\mathfrak{sl}_2(\mathbb{C})$ is a simple Lie algebra implies that every $2 \times 2$-matrix $A \in \mathbb{C}^{2\times 2}$ with $\mathrm{tr}(A) = 0$ can be expressed as a commutator of ...
0
votes
1answer
45 views

Can someone tell me books or papers on subalgebras of $\operatorname{SL}(3)$?

I hope to find the smallest subalgebra of $\operatorname{SL}(3)$ that contain the matrix $$\begin{pmatrix} 0 & a & 0\\0 & 0 & b\\c & d & 0 \end{pmatrix}$$ Are there any ...
0
votes
0answers
36 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 ...
3
votes
1answer
79 views

Inequality of Frobenius norm for skew matrices

Let $A$ be a complex skew-symmetric $n \times n$ matrix, that is, $A^T = -A$. Denote by $\|\cdot\|_F$ the Frobenius norm, that is, $\|B\|_F^2 = \text{trace}(B^*B)$. I would like to prove that $$ ...
0
votes
1answer
20 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 ...
3
votes
1answer
298 views

How to determine the matrix of adjoint representation of Lie algebra?

My questions will concern two pages: http://mathworld.wolfram.com/AdjointRepresentation.html and http://mathworld.wolfram.com/KillingForm.html In the first page, we know the basis of four matrix ...
1
vote
0answers
27 views

Nil and nilpotent restricted lie algebras

Let $k$ be a field of characteristic $p$, and let $L$ be a restricted Lie algebra over $k$. Thus $L$ is a lie algebra together with a map $(-)^{[p]}:L\to L$ satisfying the three axioms found here. ...
2
votes
1answer
58 views

What is the smallest Lie subalgebra of $ {{\frak{gl}}_{n}}(\mathbb{R}) $ whose center is the set of $ (n \times n) $-scalar matrices?

We know that the center of the Lie algebra $ {{\frak{gl}}_{n}}(\mathbb{R}) $ of all $ (n \times n) $-matrices is the Lie subalgebra of all $ (n \times n) $-scalar matrices. The Lie algebra $ ...
2
votes
1answer
117 views

When is the Cayley transform of a matrix $J$-orthogonal?

The (real) general linear group is defined $GL(n)=\{A \in \mathbb{R}^{n\times n} \mid \operatorname{det}(A) \neq 0\}$. It is a matrix Lie group. Let $J$ be a constant $n$-by-$n$ real matrix. The ...
1
vote
1answer
119 views

exponential and Lie bracket

I've been reading that one can compute a form of the Baker-Campbell formula with direct computations through the power series of $\exp(X)$, $\exp(Y)$ and $\log(1+W)$, where $X$, $Y$ are non commuting ...
3
votes
1answer
87 views

Easy proof that $\exp{Xt} = I \Rightarrow X = 0$

Let $X\in \mathbb{C}^{n\times n}$ and $I$ is identity matrix , than if: $$ \forall t\in \mathbb{R}\quad e^{Xt} = I $$ than $$ X = 0. $$ I'm looking for short and slick proof of this ...
0
votes
0answers
82 views

Which Lie group / algebra is generated by these three matrices?

This is a beginner question (and not any homework). I want to get a feeling for Lie group/algebra generators. Do the three matrices $$A=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0& ...
0
votes
1answer
76 views

anomalous minus sign in commutator of vector fields

Define $$X_A(x):=A^i_{\ j}x^j$$ where $A$ is a matrix. Why is there a minus sign in the following formula? $$[X_A,X_B]=-X_{[A,B]}$$ Edit: perhaps the question is not well posed, since what I really ...
1
vote
2answers
77 views

behavior of scalar product defined by trace under commutator

Define $$\langle X,Y \rangle := \operatorname{tr}XY^t,$$ where $X,Y$ are square matrices with real entries and $t$ denotes transpose. I have some troubles in proving that $$ \langle [X,Y],Z \rangle = ...
4
votes
1answer
55 views

solvable subalgebra

I want to show that a set $B\subset L$ is a maximal solvable subalgebra. With $L = \mathscr{o}(8,F)$, $F$ and algebraically closed field, and $\operatorname{char}(F)=0$ and $$B= ...
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 ...
3
votes
1answer
71 views

Radical of $\mathfrak{gl}_n$

I find it intuitive enough that the radical of $\mathfrak{gl}_n\mathbb F$ is the scalar matrices, but I have trouble finding an easy, but complete proof: Proof. Let $\mathfrak s$ denote the scalar ...
0
votes
1answer
193 views

Real representations of Lie algebra $\mathfrak{so}(3)$

How does one construct an $n$-dimensional, irreducible, real-valued and non-zero representation of the three generators of the Lie algebra $\mathfrak{so}(3)$ for a given value of $n$?
1
vote
1answer
173 views

A quesion in Fulton & Harris book “representation theory a first course”

In Section 11.2 A little plethysm, it discusses the tensor product of two different representations of $sl_2\mathbb{C}$. It says "If $V=\bigoplus V_{\alpha}$ and $W=\bigoplus W_{\beta}$ then ...
1
vote
1answer
69 views

Embedding of $PGL_n\mathbb{C}$ and friends

I would like the find an embedding/faithful representation from the projective linear group $PGL_n\mathbb{C}\to GL_m\mathbb{C}$ for some $m$, and likewise for the other projective groups ...
4
votes
1answer
80 views

On the ambiguity of the definition of Lie algebras of real matrix groups

I have been studying Rossmann's Lie Groups. In the context of this book, a linear group $G$ is a group of invertible real or complex matrices, and its Lie algebra $\mathfrak{g}$ consists of those ...
5
votes
1answer
277 views

Invariant Inner Product on Lie Algebra

Let $G$ be a Lie group, $\frak{g}$ its Lie algebra. Suppose $\mathcal{D}$ a representation of $G$ on $V$, $d$ the associated Lie algebra representation. Suppose $V$ is endowed with an inner product. ...
2
votes
1answer
73 views

Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
6
votes
2answers
272 views

Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
4
votes
2answers
379 views

Is every skew-adjoint matrix a commutator of two self-adjoint matrices

I'm looking to solve some matrix equations. One of the equations involves a commutator, so my question is as follows: let $A$ be a skew-self-adjoint, traceless matrix, does the equation $[X,Y] = A$ ...
1
vote
1answer
136 views

Derivative wrt. to Lie bracket.

Let $\mathbf{G}$ be a matrix Lie group, $\frak{g}$ the corresponding Lie algebra, $\widehat{\mathbf{x}} = \sum_i^m x_i G_i$ the corresponding hat-operator ($G_i$ the $i$th basis vector of the tangent ...
8
votes
2answers
365 views

Significance of the following Matrix?

I am unfamiliar with advanced Matrix theory (nor am I a mathematician), so please bear with me. Is there anything significant about the following Matrix structure? Are there any special symmetries or ...
3
votes
2answers
93 views

“How many” matrices generate 2d Lie algebra (i.e. $[c_1,c_2]=k_1 c_1 +k_2 c_2$)?

Consider a pair of matrices $(c_1, c_2)$. The words "it generates the 2-dimensional Lie algebra", means that there exists a pair of scalars $k_1$, $k_2$, such that $$[c_1, c_2] = k_1 c_1 + k_2 c_2,$$ ...
2
votes
1answer
403 views

What is the Lie algebra of the ``indefinite orthogonal group''?

For $p,q \geq 0$ and $n=p+q\geq 1$, give $\mathbb{R}^n$ the indefinite inner product (written as a matrix) $$ \begin{pmatrix} I_p & \\ & -I_q \end{pmatrix}, $$ where $I_m$ is the $m \times m$ ...
0
votes
1answer
57 views

Question about Lie algebra, where am I going wrong?

Let $M:= \begin{pmatrix}1 & 2 & 3 & 4\\3 & 4 & 4 & 1\\1 & 9 & -1 & -3\\9 &5 & -2 & -4\end{pmatrix} = \begin{pmatrix} A & B\\C & ...
2
votes
1answer
142 views

Computing the Lie algebra of $SU(n, \mathbb{C})$

The group $SU(n, \mathbb{C})$ is the set of $n \times n$ complex matrices $Q$ such that $\det Q = 1$ and $Q\overline{Q}^{T} = 1$. The Lie algebra is the set of traceless anti-Hermitian matrices. To ...
1
vote
2answers
540 views

Cayley Transform, Exponential Mapping and more…

Assume a self-adjoint operator, represented as hermitian matrix $H=H^\dagger$. To my knowledge there are at least 2 mappings of $H$ onto unitary matrices: Cayley's Transformation with ...
1
vote
2answers
351 views

“Change of basis” from standard vector space to matrix Lie algebra, and its inverse

For matrix Lie groups, the exponential map is usually defined as a mapping from the set of $n \times n$ matrices onto itself. However, sometimes it is useful to have a minimal parametrization of our ...
2
votes
1answer
435 views

Derivative of function including matrix logarithm

Is the following equation a first order approximation or incorrect for general matrix Lie groups? And what are the higher order terms? $$\frac{\partial}{\partial\mathbf x} (\log(\mathtt ...
1
vote
1answer
314 views

Closed form for the exponential of a Lie algebra 3x3 matrix?

Matrices of the form: $$\begin{pmatrix} iz+iy&-iz+w&-iy-w\\ -iz-w&iz+ix&-ix+w\\ -iy+w&-ix-w&iy+ix\end{pmatrix}$$ where $x,y,z,w$ may be assumed to be real, form a Lie ...
1
vote
1answer
185 views

Finding roots for a Lie algebra g, wrt toral subalgebra h

I'm trying to find the root space decomposition of a lie algebra wrt a toral subalgebra h. Both a matrix lie algebras. I'm confused about how do I find the linear forms $\lambda \in \mathfrak{h}^*$ ...