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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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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25 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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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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35 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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27 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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27 views

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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26 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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24 views

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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40 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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135 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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34 views

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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55 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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35 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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30 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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39 views

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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56 views

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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29 views

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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25 views

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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110 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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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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63 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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33 views

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 ...
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Showing elements of $\mathfrak{gl}_n$ can be represented as a nilpotent and a semisimple matrix under addition

I want to use the Jordan form to show that every element $A\in \mathfrak{gl}_n$ can be written as $N+S$ where $N\in\mathfrak{gl}_n$ is nilpotent and $S\in\mathfrak{gl}_n$ is semisimple and $NS=SN$. ...
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$\mathfrak{sp}_4$ is a subspace of the vector space of all $4\times 4$ matrices

Let $\mathfrak{sp}_4$ denote the set of all matrices $X$ satisfying $$X^TM+MX=0$$ How can I show that $\mathfrak{sp}_4$ is a vector subspace of the vector space of all $4\times 4$ matrices? I ...
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45 views

İf x is diagonalizable then ad(x) is also diagonalizable

I start to study lie algebras from K. Erdmann, Mark J. Wildon-Introduction to Lie Algebras and i try to solve question below but actually i can't see .How can i start ? Give me hint please Let ...
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exercise in humphrys lie algebra book

show that there exist a unique w∈ W such that wΔ=-Δ.show further that reduced expansion of w involves all simple roots.what is l(w)?
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Nested commutators that don't vanish

So I've been reading up on Lie Groups and Lie Algebras and the Baker-Campbell-Hausdorff formula. I understand how the formula works and that most of the time the nested commutators vanish at a certain ...
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41 views

Why for simple roots in Lie algebras the master formula reduces to one integer?

The master formula for two generic weights (roots) is $$ 2 \frac{\vec{a} \cdot \vec{b} }{\vec{a} \cdot \vec{a} }=q-p $$ but if we require that the roots are simple then this reduces to $$ 2 ...
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Need definition of symmetric and antisymmetric tensor representations of a Lie algebra

I couldn't find a definitive answer online. Suppose we have a representation of a Lie algebra $(\pi,V)$. Consider the symmetric and antisymmetric vector subspaces of the $k$-th tensor product of ...
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Clarification on notation of “left invariant fields” (Lie groups)

In these notes in Definition 1.4 we learn that A vector field $X$ on a Lie group $G$ is called left invariant if $d(L_g)_h(X(h))=X(g(h))$ for all $g,h \in G$, or for short $(L_g)_*(X)=X$. where ...
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Normal Subgroups of $SU(n)$

I was wondering if there is any classification for normal subgroups of $SU(n)$? In particular, I think that the answer is no for $n = 2$ by looking at the covering map onto $SO(3)$, but I was curious ...
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Help! How to derive the result related to Darboux derivative?

First, define Darboux derivative. There is one Lie group $G$ and one manifold $M$. Let $\phi:M\rightarrow G$ be a smooth map. The Darboux derivative $\Delta(\phi):TM \rightarrow M\times \mathfrak g$ ...
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Generalization of Schur's Lemma: finite dim. representations of real Lie algebras

Let $V$ be an irreducible finite dimensional real representation of a real finite dimensional Lie algebra $\mathfrak{g}$. From Schur's Lemma, what is $Hom_\mathfrak{g}(V,V)$ or ...
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58 views

Are unipotent algebraic groups connected?

Is a unipotent algebraic group over a field of characteristic zero always connected?. As far as I know, every unipotent algebraic group over field of characteristic zero is isomorphic to a closed ...
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How to prove $e^{π(ω)} = cos(ω)I_3 + (1 − cos(ω))ωˆ ⊗ ωˆ + sin(ω)π(ωˆ )$?

How to prove $e^{π(ω)} = cos(ω)I_3 + (1 − cos(ω))ωˆ ⊗ ωˆ + sin(ω)π(ωˆ )$, where $ω = |ω|, ωˆ =ω/|ω|.$ My Attempt: In my understanding, $\pi (w)$ is an element of the lie algebra, which is a ...
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Why is left-invariant vector fields needed to construct a Lie algebra from a Lie group?

Since the set of all vector fields $V$ on a Lie group $G$ forms a vector space, one can impose algebraic structure (a Lie algebra) by defining the bracket $[\cdot,\cdot]$ between these vector fields. ...
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Exterior derivatives involving representations

I have two questions regarding the exterior derivative of vector valued forms when representations are involved: Suppose $V$ is a vector space, $M$ a smooth manifold and $\omega$ is a $V$ valued ...
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The negative of a vector field and its flow

I have a relatively short question about vector fields. Let $G$ be a Lie Group, and $X$ a smooth vector field on it. If its flow is $\left\{\phi_t\right\}$ what is the corresponding flow for $-X$? ...
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Principal bundles, connection forms and fundamental vector fields

Suppose $\pi:P\rightarrow M$ is a principal bundle, $\omega\in \Omega^1(P;\mathfrak{g})$ is the connection one form and $\sigma(\cdot)$ is the fundamental vector field associated to some vector field ...
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Cohomology group of an algebra

I tried to guess what is a cohomology group of an algebra. I would like to find the correct definition of this. I know what is a cohomology group of a group, but I don't know how connect the second ...
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69 views

Is there a more intelligent way to compute the determinant of the Killing form of $\mathfrak{sl}(3,F)$?

Is there a more intelligent way to tackle exercise 7 of paragraph 5 of Humphreys (Introduction to Lie Algebras and Representation Theory)? Exercise 7: Relative to the standard basis of $L = ...
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Special linear lie algebra relative with Angular Momentum

Let have the special linear algebra $\operatorname{Sl}(2,\mathbb F)$ ,which is the set of $ 2 \times 2$ matrix with trace zero. I have to prove that the lie algebra $ g=\operatorname{Span}\{ ...
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How to visualize $\operatorname{Lie}(\operatorname{GL}_n)=\mathfrak{gl}_n$ in positive characteristic?

I'd like an intuitive explanation as to why the Lie algebra of $\operatorname{GL}_n$ is $\mathfrak{gl}_n$ when working over fields of positive characteristic. Below I reproduce how I "see" this fact ...