For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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24 views

Cartan-Maurer trivializations and Bi-invariant Metric on Lie Group

Let $G$ be a Lie group. Let $\nabla^L$ and $\nabla^R$ be the connections on $TG$ corresponding to the trivial connection $d$ on $G\times\mathfrak{g}$ under the left and right trivializations. How ...
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1answer
13 views

What really is a path-ordered exponential?

In some texts about gauge theories in Physics I've found one object called a path-ordered exponential which I'm not sure what it means. As I understood, the idea is as follows: let $G$ be a Lie group ...
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9 views

scalar multiple of Young symmetriser

The following is a lemma on Fulton and Harris' book -Representation theory,a first course (page 53): Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar ...
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0answers
24 views

Lie algebras and left or right invariant vector fields

Let $G$ be a Lie group and $\phi$ be a diffeomorphism defined by $\phi(\sigma)=\sigma^{-1} $, $\sigma\in G$. I have to prove that $X\mapsto d\phi(X)$ gives an isomorphism of algebras between the ...
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0answers
60 views
+50

Right invariance of Casimir (Laplacian)

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. The Casimir element $\Delta\in \mathfrak{zu}(\mathfrak{g})$, considered as an operator on $C^{\infty}(G)$ is right invariant, that is, $\Delta ...
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0answers
37 views
+50

At what position do we insert the new coefficient in the weights for extended Dynkin Diagrams?

Given a set of weights of a representation and the corresponding extended Dynkin diagram for some Lie algebra, we can delete a node, which yields the maximal subalgebra. I know how to draw the ...
15
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1answer
128 views

When do two matrices have the same exponential?

Let $A$ and $B$ be $n\times n$ hermitean matrices. When do we have $e^{iA}=e^{iB}$? Can we somehow classify those pairs of matrices that have the same exponential? Here are some observations that I ...
4
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4answers
1k views

Could you recommend some books on Lie algebra?

I am a pure maths student, and want to go straight ahead, so I decide to study Lie algebra on my own, and try my best to understand it from various points of view:differential equation, Lie group, ...
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1answer
8 views

$n_-$ is freely generated in a Kac moody Algebra

This question is my doubt from Kac's book on Infinite dimensional Lie algebras. We start with an arbitrary matrix A, and we define the realization of A and using the generators $\{e_i,f_i : 1 \le i ...
2
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2answers
211 views

The description of abelian Lie groups

There is a problem in my problem sheet which asks me to describe all abelian connected Lie groups (moreover this is the first problem so it should be rather easy). I don't understand how this ...
4
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0answers
15 views

Recipe to compute dimension and decompose product of $SO(N)$ group representations

As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts: There is a correspondence between irreps ...
2
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1answer
20 views

Adjoint representation and tangent vectors

Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra, $\text{Ad}:G\rightarrow GL(\mathfrak{g})$ the adjoint representation of $G$. Then, for $X,Y\in \mathfrak{g}$, \begin{align*} ...
2
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2answers
72 views

The universal enveloping algebra of free Lie algebra is the tensor algebra on the Free Abelian Group?

Let $A$ be a set of cardinality at least 2 and let $M_A$ be the free abelian group generated by $A$. Let $L(A)$ be the free Lie algebra generated by $A$. I am reading On Injective Homomorphisms For ...
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1answer
361 views

Heisenberg XXX spin model

Let $\pi$ be the standard representation of $sl_2(\mathbb{C})$ on $\mathbb{C}^2$. Let $p_1,p_2,p_3$ the three Pauli matrices. Define $S^a:=\frac{1}{2}\pi(p_a)$. What does such matrices looks like?
0
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1answer
34 views

Is adjoint map invertible?

I've already studied the group of automorphisms of a simple lie algebra on a finite field, but according to the definition of an adjoint representation of a Lie algebra, can we claim an adjoint map is ...
3
votes
1answer
38 views

Why can we write the weights of a representation in terms of the simple roots?

I'm currently trying to get my head around the fact that we can write the weights of any representation in terms of the simple roots of the algebra. Is there any, not too-technical, explanation? I ...
2
votes
1answer
93 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 ...
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1answer
21 views

Finite dimensional representations of semi-simple Lie algebras

I've been trying to understand the proof of the following statement: An injective map of $\mathfrak{g}$-representations of a semisimple Lie algebra splits. I'm supposed to show this considering the ...
0
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1answer
21 views

L-module definition

I have the following definition of an L-module We say that V is an L-module if there is a k-bilinear mapping L × V → V sending a pair (x, v) ∈ L × V to x.v ∈ V such that [x, y].v = x.(y.v) − y.(x.v) ...
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0answers
18 views

Complex irreducible representation of solvable lie algebra

How can one infer from the Lie's theorem (in terms of existence of a common eigenvector) that a complex irreducible representation of a solvable lie algebra has dimension 1? What I know is that one ...
1
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1answer
28 views

Perfect Lie algebras

How can I prove that $gl(n,k)$ and $sl(n,k)$ with $[x,y]=xy-yx$ are perfect algebras? By definition ,$g$ is a perfect algebra if $g=g\prime$, where $g\prime=<\{[x,y]| x,y\in g\}.$
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0answers
25 views

How to prove the Lie bracket is infinitesimal commutator

I am currently studying Lie groups and I cannot solve the following exercise, which I think is vital to my understanding. The Lie bracket is defined as $[X,Y]=\text{ad}(X)Y$. Let the group commutator ...
3
votes
1answer
22 views

What are some good invariants for low dimensional Lie algebras?

I'm working out some computations on Lie algebras $L$ of low dimensions (by which I mean $3, 4$ or $5$). For my purposes, it is convenient to choose an orthonormal basis $\{e_1, e_2, \ldots, e_n\}$, ...
0
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1answer
20 views

Is decomposition of a semisimple Lie algebra unique?

A semisimple Lie algebra is defined to be the sum of simple Lie algebras. But is this decomposition to simple Lie algebras unique? If not can you give an example?
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1answer
36 views

Killing forms and Hermitian inner products

Let $K$ be a compact, connected, simply connected Lie group with Lie algebra $\mathfrak k$ and Killing from $B_{\mathfrak k}$. It is well known that $B_{\mathfrak k}$ is a negative definite symmetric ...
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1answer
581 views

Computation of the Killing form of $\mathfrak{gl}_{m}$.

Consider the Killing form of the Lie algebra $\mathfrak{gl}_{m}$. Then $\{e_{ij}\}$ is a basis for this Lie algebra where $e_{ij}$ is a matrix with 1 in the $i$th row, $j$th column and 0 everywhere ...
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0answers
20 views

Subspaces of Lie algebras

The Lie correspondence is well understood. For 'nice enough' Lie groups $G$ (with Lie algebra $\mathfrak{g}$) every sub-group $H < G$ has a Lie algebra $\mathfrak{h} < \mathfrak{g}$ given by ...
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1answer
44 views

How can I maintain linear independence through a commutator?

Consider a Lie algebra $\mathcal{L}$, a linearly independent generating set $\mathcal{G}$, and an element $X \in \mathcal{L}$. Edit: Note that $\mathcal{G}$ is not necessarily a basis; the generation ...
12
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1answer
143 views

Linear Algebra : Invertible Matrix Proof

I was doing some linear algebra exercises and came across the following tough problem : Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose ...
7
votes
2answers
479 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
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0answers
33 views

Which elements of $su(n)$ commute with those of a subalgebra $su(2)$

Given a subalgebra $su(2) \subset su(n)$ , how many generators of $su(n)$ commute with any element in the subalgebra $su(2)$? I know that there are at least $n-2$ elements in $su(n)$ satisfying this ...
3
votes
2answers
62 views

Not null Killing form.

I have to find an example of solvable Lie algebra $L$ such that the Killing form of $L$ isn't null. If we take the Borel subalgebra of $\mathfrak{sl}(2)$, we have that the Killing form of $L$ is the ...
3
votes
2answers
81 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 ...
3
votes
1answer
49 views

What intuition do we have for a subalgebra of Lie to be abelian?

The motivation for my question comes from the definition of rank of a given globally symmetric space: it is based on the image of a maximal abelian subalgebra of a given algebra by the exponential ...
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1answer
36 views

Weights in $\mathfrak{sl}(3,\mathbb{C})$

Let $\mathfrak{h} \subset \mathfrak{sl}(3,\mathbb{C})$ be the set of diagonal matrices. Then for $A = \begin{pmatrix} a_1 & 0 & 0 \\ 0 & a_2 & 0 \\ 0 & 0 & a_3 \end{pmatrix} ...
0
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2answers
18 views

Lie algebra, maximal toral sub-algebra

Is there a relation between number of roots of a finite dimensional semi-simple Lie algebra L and dimension of the maximal toral sub-algebra H(Cartan sub-algebra) of L? Thanks!
2
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0answers
76 views

Differential operators on the polynomial ring

Let $A$ be a commutative algebra over complex numbers. If $a\in A$ we define $m_a$ to be a linear map which sends each $x$ to $ax$. The zero map $A\to A$ is said to be a differential operator of an ...
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1answer
56 views

General differentials operators (Grothendieck definition) and polynomial rings

Let $A$ be an algebra over some field $\mathbb{k}$. A linear map $f:A\to A$ is said to be a differential operator of an order $\le n$ if for all $a_0,a_1,\ldots a_n\in A$ we have ...
0
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0answers
33 views

Can the adjoint representation on a discrete centered group merge connected components?

Let $\mathfrak{G}$ be a Lie group with algebra $\mathfrak{g}$ and with in general many connected components but which is either centerless or has at most a discrete center. Then we have an ...
0
votes
1answer
22 views

Can we represent the curl as a multiplication by skew-symmetric matrix?

Considering that two vectors $A \times B$ = $\hat A* B$, where $\hat A$ is a skew symmetric matrix containing elements of $A$ Can we then write the curl $\nabla \times A$ as $\partial \vec r *A$ ...
3
votes
2answers
37 views

How to descent to smaller groups “by chopping off a node of the Dynkin diagram”?

I read in section 2 of this paper : "There is a well-defined chain to descent from $E_8$ to smaller groups by chopping off a node of the Dynkin diagram." What exactly is here referring to ...
0
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1answer
16 views

Killing form of a reductive symmetric Lie algebra

suppose $(g; , k ,p)$ is a reductive symmetric Lie algebra. i.e. $k$ is a sub-algebra of $g$, $[k,p] \subset p$ , $[p,p] \subset k$ and $g= k \oplus p$. this is actually from Lepowsky and McCllum's ...
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1answer
25 views

Why Demazure operator is an endomorphism of $\mathbb{Z}[P]$?

Let $P$ be the weight lattice of some Lie algebra. Let $$ \Delta_{\alpha}(u) = \frac{u-s_{\alpha}\cdot u}{1-e^{-\alpha}}, $$ where $\alpha$ is a root, $u \in P$. In the article, it is said that ...
3
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1answer
30 views

Why not every homogeneous manifold is parallelizable?

It is obvious that not every homogeneous manifold is parallelizable (take for example the two-sphere $S^{2}$). In contrast, every Lie group $G$ is parallelizable, as you can construct a pointwise ...
2
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1answer
19 views

Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$?

I just watched this lecture and there Susskind says that $${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$ for the Lie group $G= SU(N)$. Unfortunately, he does ...
2
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0answers
27 views

Write down the explicit form of the $15$ Killing vectors of the 5-sphere

I am looking for a way to write down explicitly the $15$ vectors which are generators of $SO(6)$ in polar coordinates on the $5$-sphere. In particular I have the round metric $$g_{\mu\nu} = \left( ...
2
votes
2answers
62 views

Showing a linear combination of matrices is nilpotent for any constants

So I have three linear operators in a $3$-dimensional vector space $V$ over field $\Bbb k$ whose matrices w.r.t basis of $V$ are $$X= \left(\begin{matrix}1 & 0 & 1\\ 1 & 0 & 1\\ -2 ...
0
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1answer
44 views

finding eigenvalues and eigenspaces of a linear operator

Assume that $char(\mathbb{k}) = p > 3$ and let $W(1)$ be the Witt algebra over $\mathbb{k}$. Recall that $W(1) = Der(A)$ where $A = k[t]/(t^p)$, a truncated polynomial ring over $\mathbb{k}$. ...
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1answer
60 views

direct sum and tensor product of representation of lie algebra

Let $(p_1,V_1)$ , $(p_2,V_2)$ representation of a lie algebra $g$ on $V_1,V_2$. I have to prove that: $ i) $ the direct sum $p_1 \oplus p_2$ is a representation of $g$ in $V_1 \oplus V_2$ $ ii) $ ...
2
votes
0answers
31 views

How to write $\mathfrak{su}(3)$ Lie algebra as a sum of two subspaces?

Let $K,F\subset\mathfrak{su}(3)$ be subspaces, such that $K \oplus F =\mathfrak{su}(3)$, and $K$ has a $\mathfrak{su}(2)$ structure. How can we show that $[K,K] = K$ (i.e., commutator of any two ...