6
votes
0answers
42 views
+50

Invariant tensors in adjoint representation

Suppose we have a simple Lie group $G$ with algebra $\mathfrak{g}=\{X_a\}$, where the generators $X_a$ are in some matrix representation. Is it true that the only invariant rank $n$ tensor in the ...
1
vote
0answers
31 views

Derived series of a Lie algebra

I've been studying semisimple Lie algebras and solvability and was wondering if someone could explain to me the meaning of the derived series of a Lie algebra L and this part: $$L^{(1)}=[LL]$$ I don't ...
2
votes
0answers
18 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
37 views

If HP=PH+P for H,P n×n complex matrices, must H be diagonalizable?

If $F$ is a field of characteristic zero, $H,P$ are $n\times n$ matrices over $F$, $0 \neq \alpha \in F$, and $HP=PH+\alpha P$, then must the minimal polynomial of $H$ be square-free and must $P$ ...
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
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:= ...
2
votes
1answer
43 views

Is every element of a complex semisimple Lie algebra a commutator?

Let $L$ be a (finite-dimensional) complex semisimple Lie algebra. Then we know that $L = [L,L]$. Is it true that every element of $L$ must be a commutator? Since a complex semisimple Lie algebra is ...
5
votes
2answers
68 views

$\mathfrak{so}(n)$ has trivial center when $n\geq 3$

Is there a nice way to show that $$\mathfrak{so}(n)=\{A \in M(n,\mathbb{R}): A+A^t=0\} $$ has zero center for $n \geq 3$?
1
vote
1answer
43 views

The trace as an integral over a sphere [duplicate]

Let $V$ be a real vector space of dimension $n$ and let $\langle \, \cdot\, , \,\cdot\, \rangle$ be an inner product on $V$. We can define a linear functional on the space of endomorphisms of $V$ by ...
0
votes
0answers
30 views

Infinite series of nested commutators

I'm trying to show the following: If $S_i$ are a set of three matrices such that $$ [S_i, S_j] = \epsilon_{ijk} S_k $$ then $$\exp\big( \alpha_i [S_i, \cdot]\big) S_j = (\exp (M) \vec{S})_j$$ ...
5
votes
2answers
124 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 ...
4
votes
3answers
142 views

How to prove that $B^\vee$ is a base for coroots?

Let $\Phi$ be a root system in a real inner product space $E$. Define $\alpha^\vee = \frac{2\alpha}{(\alpha, \alpha)}$. Then the set $\Phi^\vee = \{\alpha^\vee: \alpha \in \Phi \}$ is also a root ...
0
votes
1answer
54 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 ...
0
votes
0answers
36 views

Basis of Witt algebra

The Witt algebra $W(n,m)$ is defined as the set of element $\{\sum f_j D_j$ such that $ f_j ∈ A(n,m)\}$ with usual Lie bracket. I am a bit confused about basis for $W(n,m)$? What is the meaning of ...
1
vote
1answer
18 views

Uniqueness of the Lie brackets in the quotient space of a Lie algebra

Suppose I have a Lie algebra $\mathfrak g$ which is an ideal of $\mathfrak a$. Then I consider the quotient set $\mathfrak g / \mathfrak a$ which is the set of all equivalence relations of $\mathfrak ...
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
63 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
0
votes
1answer
20 views

Show that $(X^*)^*=\epsilon\epsilon 'X$

A finite dimensional vector space $V$ with a non-degenerate form (,) s.t. $(u,v)=\epsilon (v,u) \forall u,v\in V$ is called a quadratic space of type $\epsilon$. Let $V$ be a quadratic space of type ...
1
vote
1answer
36 views

Why $X^* X\in \mathfrak{g}(V)$.

A finite dimensional vector space $V$ with a non-degenerate form (,) s.t. $(u,v)=\epsilon (v,u) \forall u,v\in V$ is called a quadratic space of type $\epsilon$. Let $V$ be a quadratic space of type ...
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
128 views

Jordan–Chevalley decomposition

I'm trying to understand the proof of it in Humphreys(Humphreys 1972, Prop. 4.2, p. 17). And I've not got over which field we are working. The characteristic polynomial may not have roots in the ...
4
votes
1answer
78 views

Perturbation in characteristic p, or Why, really, does Lie's theorem fail?

While recalling some basics of Lie theory, I found a funny proof of the main lemma in Lie's theorem on triangularity of representations of solvable Lie algebras. It turns out that this proof has a ...
4
votes
1answer
57 views

Epimorpsims preserve generalized eigenspaces

This is most likely trivial, but I don't get it. In Humphrey's Introduction to Lie algebras, page 82, he says: It is clear that, if $\phi \colon L \to L'$ is epimorphism [of finite dim. Lie ...
3
votes
1answer
59 views

Automorphism in the special linear algebra $\mathfrak{sl}_2(F)$

If $L=\mathfrak{sl}(n,F), g\in GL(n,F)$, prove that the map of $L$ to itself defined by $x\rightarrow -gx^tg^{-1}$ ($x^t=$transpose of $x$) belongs to $\operatorname{Aut}L$. When $n=2,g=$identity ...
5
votes
1answer
127 views

Subspace spanned by eigenvectors is a subalgebra

Let $L$ be a Lie algebra over an algebraically closed field and let $x\in L$. Prove that the subspace of $L$ spanned by the eigenvectors of $\operatorname{ad}x$ is a subalgebra. Suppose the ...
3
votes
1answer
88 views

Eigenvalues of $\operatorname{ad}x$

Let $x\in \operatorname{gl}(n,F)$ have $n$ distinct eigenvalues $a_1,\ldots,a_n$ in $F$. Prove that the eigenvalues of $\text{ad }x$ are precisely the $n^2$ scalars $a_i-a_j$ ($1\leq i,j\leq n$), ...
3
votes
2answers
83 views

Equivalent definitions of Verma modules

This is a rather basic question. I was reading some notes on geometric representation theory by Gaitsgory and his defition of Verma module is the following: Let $ \lambda $ be a weight of $ ...
2
votes
0answers
113 views

Root-subspaces are $\mathfrak{g}$-invariant

I need some help with technicalities concernig root-subspaces of nilpotent Lie algebras of operators. Let $\mathfrak{g} \subseteq \mathfrak{gl}(V)$ be nilpotent Lie algebra, $\alpha \ \colon \ ...
0
votes
2answers
75 views

How to work out the inverse matrix $A^{-1}$ ?

Suppose A is a matrix over some ring R (might be non-commutative). How to work out the inverse matrix $A^{-1}$?
6
votes
2answers
85 views

$V=\{A\in M_{3\times 3}(\mathbb{R}):\text{trace}(A)=0\}$ is isomorphic to $\text{span}\{AB-BA:A,B\in V\}$

Background: Let $$V:=\{A\in M_{3\times 3}(\mathbb{R}):\text{trace}(A)=0\}$$ be the vector space of $3\times 3$ real matrices with vanishing trace, and let $[\cdot,\cdot]:V\times V\to V$ be defined by ...
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 ...
2
votes
0answers
66 views

Is algebraic closure required in Weyl's theorem on complete reducibility? (Lie algebras)

Weyl's theorem states that finite-dimensional representations of finite dimensional semisimple Lie algebras are completely reducible (expressible as a direct sum of irreducible submodules), with some ...
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
112 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 $M(n, \mathbb{R})$.
6
votes
2answers
615 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 ...
0
votes
1answer
100 views

Orthogonal subspace relative to the Killing form

I'm following a book in Humphrey's Introdutction to Lie Algebra's and Representation Theory. I'm reading the proof that a semisimple Lie algebra is the direct sum of simple modules. It uses the ...
4
votes
2answers
367 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 $[X,Y] = A$ ...
0
votes
1answer
50 views

preserves eigen spaces?

"Let $H_0=\begin{pmatrix}i&0\\0&-i\end{pmatrix}$, suppose $A\in SU(2)$ commutes with $H_0$, it must preserves each eigen spaces for $H_0$, eigen spaces for $H_0$ are just $\mathbb{C}e_1$ and ...
6
votes
1answer
86 views

Question about 8.4 in Humphreys

I am reading section 8.4 in Humphreys' book Introduction to Lie Algebras and Representation Theory. He is showing that the only scalar multiples of a root are 1 and -1, but I have trouble ...
4
votes
2answers
147 views

If $\exp(itH) A \exp(-itH) = A$ for all $t$, do $A$ and $H$ commute?

Let $H$ be a self-adjoint $n \times n$ matrix with complex entries. $H$ gives rise to a continuous 1-parameter group of unitaries $t \mapsto U_t = \exp(itH) : \mathbb{R} \to U(n)$. Let $A$ be some ...
0
votes
1answer
429 views

Vector space generated by the tensor products of pauli matrices

Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices: \begin{equation} ...
5
votes
1answer
117 views

What is good about simple Lie algebras?

Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple. I wonder what beautiful properties does a ...
2
votes
1answer
394 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 $m \times m$ ...
9
votes
1answer
351 views

Inscrutable proof in Humphrey's book on Lie algebras and representations

This is a question pertaining to Humphrey's Introduction to Lie Algebras and Representation Theory Is there an explanation of the lemma in §4.3-Cartan's Criterion? I understand the proof given there ...
0
votes
0answers
122 views

Derivative/Chain Rule (for MANLYfolds) Computation

Embarrasingly, I can't compute the following derivative. $dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}$, where $X$ resides in the lie algebra of $\rm SL(3,\Bbb C)$ [ie $\mathfrak{sl}(3,\Bbb C)$] ...
3
votes
2answers
413 views

Lie Algebra Homomorphism Question

So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
1
vote
1answer
70 views

Understanding a statement about the existence of functionals

Thm. Let $L$ be a solvable subalgebra of $gl(V)$, $V$ a finite dimensional nonzero vector space. Then $V$ contains a common eigenvector for all the endomorphisms in $L$. The proof of this theorem is ...
3
votes
1answer
132 views

What is the difference between $\ker( L \bigwedge L \overset{[-,-]}{\rightarrow} L )$ and $\langle a \wedge b \big| [a,b]=0\rangle$?

Let $L$ be a finite dimensional Lie algebra. We view the Lie bracket as a linear map on the exterior square: $$\pi:L \bigwedge L \rightarrow L$$ Define $$\bigwedge L := \langle a \wedge b \big| ...
6
votes
1answer
134 views

Kernel of the Lie bracket

Let $\mathfrak{g}$ be a dimension 3 Lie algebra and $[\quad,\quad]$ be a rank 1 map from $\bigwedge^{2}\mathfrak{g} \rightarrow \mathfrak{g}$. In this case, the kernel of $[\quad,\quad]$ is $3 - 1 = ...