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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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!
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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 ...
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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 $\partial \vec r *A$ ...
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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 ...
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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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Proof theorem of Lie's Algebra [on hold]

I need help with a proof this theorems, anyone could be help? I need a proof this theorems. Corollary of Lie's Theorem: Let $L$ be a solvable subalgebra of $\text{gl}(V)$, $\dim V = n$ (finite). ...
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Dimension of adjoint orbits in $\mathfrak{su}(n)$

What is the dimension of the sub-manifold $M(A)$ of $\mathfrak{su}(n)$ defined by: $M(A) = \{U^{\dagger} A U \ \text{s.t.} \ U \in SU(n) \}$ for each $A \in \mathfrak{su}(n)$.
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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 ...
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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( ...
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1answer
24 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 ...
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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 ...
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55 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) $ ...
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8 views

Commutation Relation - Generators of Semisimple Lie Group

The following is stated in a book I picked up, Group Theory for High-Energy Physicists - M. Saleem, M. Rafique. Consider an $r$-parameter semisimple Lie group of rank $\ell$. It has a set of ...
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42 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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55 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 ...
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29 views

Is there a general method to calculate the generators of the subgroups of $\textrm{GL}(n,F)$?

I know this might be a very bad/broad question, but after going through a few practice problems for finding linearly independent generators for some of the easier subgroups of $\textrm{GL}(n,F)$ ...
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33 views

Finding lower central series of Lie algebra L

I have a worked example of finding the lower central series but can't get my head around it. So L= Hn, the nth Heisenberg Lie Algebra. Then L has basis: {u1,...,un, v1,...vn, z} and: [ui, uj]=[vi, ...
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Showing that the Witt algebra $W(1)$ is isomorphic to $\mathfrak{sl}(2,\mathbb{k})$

As the title suggests, I need to show that the Witt algebra $W(1)$ with basis $\{e_i \, | \, -1 \leq i \leq p-2\}$ where $e_k=t^{k+1}\frac{\mathrm{d}}{\mathrm{d}t}$ with Lie bracket defined by ...
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1answer
27 views

calculating Jacobi identity with 5 elements in basis

I have been given an anticommutative $\mathbb{k}$-algebra $L$ with basis $\{a,b,c,d,e\}$ . I need to verify that $L$ is a Lie algebra, i.e the Jacobi identity $=0$ for any three elements $\in L$. My ...
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What is the fastest way to calculate the character of a fundamental representation?

Suppose we know the Cartan matrix $C$ of a Kac-Moody algebra $\mathbb g$: $$ C=\left(\begin{array}{cccc} 2 & -1 & 0 & 0\\ -1 & 2 & -1 & -1\\ 0 & -1 & 2 & 0\\ 0 ...
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1answer
27 views

How should I build a SU(4) matirx with a C4 vector?

I have a complex vector $S=[S_1,S_2,S_3,S_4]$ with $|S_1|^2+|S_2|^2+|S_3|^2+|S_4|^2=1$. My question is how to bulid a matix $C\in SU(4)$ while \begin{equation}C= \left( \begin{array}{cccccc} S_1 ...
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25 views

Cartan's Criterion for Solvability

I'm trying to understand the proof of Cartan's Criterion for Solvability given here, and have two questions: On page 15, about half way down, we assert the following: If $\mathfrak{g}=\mathfrak{g}_0 ...
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21 views

Map to the submodule of invariants of a Lie algebra representation

If $G$ is a compact group and $V$ is a representation, the inclusion $V^G \to V$ has an easy-to-write-down retract: \begin{equation*} V \to V^G,\:\: v \mapsto \frac{1}{|G|} \int_G g\cdot v\;dg ...
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21 views

How to write su(3) Lie algebra as a sum of two subspaces? [duplicate]

Let K,F⊂su(3) be subspaces, such that K⊕F=su(3), and K has a su(2) structure. How can we show that [K,K]=K (i.e., commutator of any two elements of K gives an element in K), [K,F]=F, and [F,F]=K?
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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 ...
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1answer
18 views

Dimension of set of Hermitian matrices commuting with a given matrix

Given a Hermitian matrix $A$, what is the dimension of the set of all other Hermitian matrices $B$ such that $[A,B] = 0$. It is clearly not the same for all $A$, but how can one find it for a given ...
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check that $x,y,z$ span a 3-dimensional Lie subalgebra $L$ of $\mathbb{gl}(V)$

Suppose $V$ is a 3-D vector space over a field $k$ with basis $B=\{v_1,v_2, v_3\}$ and consider the linear operators $x,y,z\in\mathbb{gl}(V)$ whose matrices with respect to $B$ are some 3 by 3 $X, Y$ ...
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Why generator in Lie Algebra is defined as the coefficient in taylor expansion of map

Booth defines the infinitesimal generator of a lie group (denote the manifold it defines by $M$) using flow $\theta_t(p)$ by calculatng the limit (mainly the derivation for $f$ in each point $p\in M$) ...
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1answer
22 views

Example of a semisimple Lie algebra with degenerate Killing form

We know that when the killing form of a Lie algebra is nondegenerate then it is semisimple. I am looking for a semisimple Lie algebra with degenerate killing form. I know if the field is of ...
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1answer
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Showing that the universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is isomorphic to its opposite ring

One of my lecturers mentioned in passing that the universal enveloping algebra of a Lie algebra is isomorphic to its opposite ring, so I wanted to prove this fact. To this end, let $\mathfrak{g}$ be ...
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Lie algebra of a lie group

In proposition 5.2 of Bump's Lie Groups, he states: Let $G$ be a closed Lie subgroup of $GL(n, \mathbb{C})$, and let $X \in Mat_n (\mathbb{C})$. Then the path $t \to exp(tX)$ is tangent to the ...
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1answer
20 views

basis for symmetric square

If we have the symmetric square Sym$^{2}V$ and $V=\mathbb{C}^{2}$, why is it that $\{x^{2}, xy, y^{2}\}$ form a basis for it? So symmetric square matrices are when the main diagonal acts as a ...
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adjoint of projection operator

|$\phi_m$> are the eigenstates of a Hermitian operator H. Assume that the states |$\phi_m$> form a discrete orthonormal basis. The operator U(m,n) is defined by: U(m,n) = ...
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37 views

Determining the Derived Series and the Lower Central Series of a Lie Algebra

Suppose we have a Lie algebra $L$ over $\textbf{k}$ with basis $\{x,y,z\}$ and with $$[x,y]=z, [y,z]=x, [z,x]=y.$$ How do I go about finding the lower central series and the derived series for $L$? ...
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119 views

Why are Lie algebras “rigid” objects?

I read the following motivation for quantum groups on wikipedia: The discovery of quantum groups was quite unexpected, since it was known for a long time that compact groups and semisimple Lie ...
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Lie Algebra associated to a lie group [closed]

Given an infinite dimension vector space, let $G=I+End^f(V)$ where $End^f(V)$ is the ideal of finite rank endomorphism, and $H=G_1\subset G$ of endomorphisme of determinant $1$. Could you help me ...
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44 views

what is the algebra generated by a set of matrices?

this is from wikipedia unde "A set of matrices $A_1, \ldots, A_k$ are said to be simultaneously triangularisable if there is a basis under which they are all upper triangular; equivalently, if they ...
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28 views

meaning of conjugate Cartan subalgebras

what does it mean that all Cartan subalgebras are conjugate under automorphisms of the Lie algebra if the field is algebraically closed?
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28 views

Group exponentials and general group of diffeomorphisms

I read on the wiki page (http://en.wikipedia.org/wiki/Exponential_map_%28Lie_theory%29) that the group exponential is not a local diffeomorphism at all points. Can someone give me an example?
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Dual of a matrix lie algebra

In fact I already calculate the dual space with a formula, but I did'd understand some steps of the formula. So, I want to calculate the dual space of The lie algebra of $SL(2,R)$. Knowing that ...
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Semisimple implies complete reducibility

Why does a semisimple Lie algebra imply complete reducibility? I have that a semisimple Lie algebra is a Lie algebra with no non-zero solvable ideals. Complete reducibility means that every invariant ...
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Simple algebra over algebraically closed field

In Jacobson's Lie Algebras, page 303, it seems he uses the following result: If $\mathfrak L$ is a simple finite-dimensional Lie algebra over a field $\Omega$ which is the algebraic closure of a ...
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A doubt from the Isomorphism theorems of Lie algebras.

Given an isomorphism of two irreducible root systems $\Phi$ and $\Phi$' we need to show that the corresponding simple Lie algebras $L$ and $L'$ are isomorphic. For that we take the subalgebra $D$ of $ ...
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26 views

Adjoint of Exponential Map

If $\exp: T_p(G) \rightarrow G$ is the expoenential map of a lie group, then what does the adjoint operator (as in $\langle Ax,y\rangle=\langle x,A^*,y\rangle$) of the derivative of exp look like? ...
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Are compact Lie algebras necessarily compact as a set of matrices?

I'm reading through a paper and came across something confusing; my limited experience with Lie theory is a bit of a hindrance: The author starts with a compact set of matrices (in the usual ...
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What information can I immediately extract from a Dynkin diagram?

I have understood quite well how we construct Dynkin diagrams. My question is the following: What immediate information can I extract just by looking at a Dynkin diagram? Of course I can ...
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Metrics on affine connections

In one of the paper's I read this statement: "the affine geodesics of the Cartan connections (group geodesics) are metric-free". What does this really mean? paper: ...
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53 views

Algebra is Generated by nilpotent Lie Algebra

$X$ Banach space. In $B(X)$, we can define a Lie product $[ , ]:[T_1,T_2]=T_1T_2-T_2T_1$ for any $T_1,T_2 \in B(X).$ Let $\mathcal{L}$ Lie Algebra. $\mathcal{L}^1=\mathcal{L}$ , $ ...
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1answer
53 views

Intuition for the exponential of a matrix

I'm trying to understand an algorithm that tries to map points from a lie group to its lie algebra using the exponential map. The background is the representation of 3d coordinate transformations as a ...
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Lie bracket simplification

Could someone help me simplify the following: Let $$X= -x^1\frac{\partial}{\partial x^1}+x^2\frac{\partial}{\partial x^2} \qquad Y = x^2\frac{\partial}{\partial x^1}$$ Calculate $[X,Y]$ This ...