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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Show that $\pi(Z)$ acts as a scalar over $\mathbb{g}$

Let $(\pi, V)$ be a finite dimensional irreducible representation of $\mathbb{g}$ $V$ is a vector space of homogeneous polynomials in 3 variables of degree d over $\mathbb{R}$ ...
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9 views

Generate Lie-Algebra $su(N)$

Let $A \in \mathbb{C}^{n \times n}$ be a diagonal matrix with $Tr(A)=0$ and all eigenvalues (purely imaginary eigenvalues) are mutually different from each other. Hence, in particular $A \in ...
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1answer
49 views

Relations between center (fundamental group) and (co)root and weight lattices for Lie groups

I would like to find some explanation or reference for the following facts, provided they are correct, and clarify some of the assumptions. Denote by $G$ a (perhaps semisimple compact connected) Lie ...
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1answer
24 views

Sextonion Cayley Table

I've been reading up on the sextonions and was wondering if it would be possible to construct a Cayley table for the split sextonions the same way as one would do so for the split quaternions and ...
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1answer
22 views

Split Lie algebra extensions?

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras. A Lie algebra extension is a short exact sequence $$0\longrightarrow \mathfrak{h}\stackrel{\jmath}{\longrightarrow} ...
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1answer
39 views

Why is 1/2+1/2 in the weight space for SO(5)

Let's consider $\mathfrak{so}(5)$ as the Lie algebra of $\mathrm{SO}(5)$, where the symmetric bilinear form is $x_1y_5+\cdots +y_1x_5$. Then the maximal torus is given by $$\left(\begin{array}{cccccc} ...
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1answer
46 views

Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
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1answer
22 views

Every irreducible representation of $G_2$ appears in some tensor power of the standard representation

In the Book "Representation Theory" by Fulton and Harris, this fact ist stated on page 353 after looking at the weight diagrams of the complex Lie-Algebra $G_2$. The authors deduce that with ...
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1answer
21 views

Functoriality of the adjoint representation

Just a simply question. I came across the following statement which is used for deriving Weyl's integral formula: ''$\text{Ad}_G(h)|_{\mathfrak{h}} = \text{Ad}_H(h)$ due to functoriality in the Lie ...
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Field of char(F)=p, if p divides n then sl(n,F) is not simple [on hold]

Self-Studying Let $L$ be a finite-dimensional complex Lie algebra and let $\alpha$ be an automorphism of $L$. For $ν\in C$, let $L_ν =\{x\in L:\alpha(x)=ν(x)\}.$ Show that $[L_\lambda, L_\mu] ⊆ ...
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Using Serre's Theorem, and showing lie algebra is 1-D [on hold]

Help! Self-studying Use Serre’s Theorem to show that the Lie algebra $so(6, C)$ is isomorphic to $sl(4, C)$, where $C$ denotes the complex numbers. Let $L$ be the Lie algebra generated by $x, y, z$ ...
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16 views

Dimensions of classical lie algebras and showing $sp(4,C) \cong so(5,C)$ [on hold]

Please help me with proofs for these two questions. They are for my own studying and I have no idea where to start. Show that the dimensions of the classical Lie algebras are as follows: ...
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0answers
22 views

Roots Systems, Dynkin Diagrams and more help! [on hold]

Hi I am independently studying Lie Algebras and I am stuck on these two problems from a book. Anything would help! Thanks! Show that the root system is irreducible if and only if its Dynkin diagram ...
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1answer
55 views

Left Invariant vector field on SO(3)

I have a Lie group, namely on SO(3), i.e. $SO(3,\mathbb{\mathbb{R}})=\left\{ A\in GL\left(3,\mathbb{R}\right)\mid A^{T}A=\mathbb{1},\,\det\left(A\right)=1\right\}$. I have a Left action $L_g$ and I ...
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2answers
27 views

Angular momentum operators

Suppose we have angular momentum operators $L_1,L_2,L_3$ which satisfy $[L_1,L_2]=iL_3$, $[L_2,L_3]=iL_1$ and $[L_3,L_1]=iL_2$. We can show that the operator $L^2:=L_1^2+L_2^2+L_3^2$ commutes with ...
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1answer
61 views

Lie rings: reference request

Dear friends: I am looking for a modern reference for Lie rings (In particular, I would like to have nice references for the structure of Lie ideals), let it be lecture notes or a book, in the sense ...
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2answers
26 views

Bruhat decomposition and the order of the group.

Suppose we have a group $G(q)$ over a finite field $\mathbb{F}_q$. How can the Bruhat decomposition be used in order to calculate the order of $G(q)$? Are there any examples for some particular ...
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14 views

$\mathfrak{sl}_2$ has the root lattice of type $A_1$.

Let $L$ be a Lie algebra over $\mathbb{Z}$ constructed from a root lattice $R$. It is well-known that if $R=A_1$, then $L \cong \mathfrak{sl}_2$ and this is widely used example in many books on Lie ...
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0answers
29 views

Finite dimensional algebraic representation of $SL_2(\mathbb{C})$

I heard that for each $n\in \mathbb{N}$, there is the unique algebraic irreducible representation of $SL_2(\mathbb{C})$ with dimension $n$ over $\mathbb{C}$. Would you let me know what is such ...
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1answer
28 views

$ [[[A,B],C],D] + [[[B,C],D],A] + [[[C,D],A],B] + [[[D,A],B],C] = 0 $

If A and B are $n \times n$ matrices, define the Lie product $[A,B] = AB-BA$. Exercise 1.37 of the book Basic Linear Algebra by T.S. Blyth and E.F. Robertson asks to prove that $$ (*) \ \ \ \ \ ...
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1answer
58 views

Prove that the sum of all simple roots is a root

Let $\Delta$ be an indecomposable root system in a real inner product space $E$, and suppose that $\Phi$ is a simple system of roots in $\Delta$, with respect to an ordering of $E$. If $\Phi = ...
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2answers
840 views

Calculating the Lie algebra of $SO(2,1)$

I am trying to calculate the Lie algebra of the group $SO(2,1)$, realized as $$SO(2,1)=\{X\in \operatorname{Mat}_3(\mathbb{R}) \,|\, X^t\eta X=\eta, \det(X)=1\},$$ where $$\eta = \left ( ...
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0answers
27 views

Non-Lie character of Leibniz algebra

Let $J$ be the largest ideal of Leibniz algebra $L$ which denotes the non-Lie character of $L$. Is it possible to write $L=L_{Lie}\cap J$? We know that $L_{Lie}= L/J$. I am going to give the following ...
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1answer
14 views

Show that we have a smooth path in $T_1(G)$, the tangent space of a matrix group

Consider the path $D_s(T)=A(s)B(t)A(s)^{-1}B(t)^{-1}$ in $G$ for some fixed value of s. Then the Lie bracket $[X,Y]$ can be related to the commutator of $A(s)B(t)A(s)^{-1}B(t)^{-1}$ of smooth paths ...
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0answers
40 views

Irreducible Representation of $sl(3,C)$

We know that the the roots of $\mathbb{g} = sl(3,C)$ under the adjoint action are given by $L_i - L_j$ where $L_i (diag(a_1, a_2, a_3))=a_i$ for $i = 1,2,3$. If $V$ is any irreducible representation ...
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1answer
16 views

Isotropy algebra for $U(n)$? [duplicate]

Let $G = U(n)$ be the Liegroup of $n \times n$ unitary matrices and $\mathfrak{g}$ the corresponding Lie algebra. Now $G$ can act on $\mathfrak{g}$ by the Adjoint-action. Since $G$ is a subgroup of ...
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3answers
208 views

Prerequisites to “Applications of Lie Groups to Differential Equations”

I'm currently a 4th year student at a university. I've become close with a professor and we talked about the topic of lie groups in differential equations. He then offered to do a reading course with ...
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1answer
47 views

Infinite Lie algebra

It is well known fact that the finite dimensional Lie algebra will always be closed under commutation relation. But I have doubt about infinite Lie algebra. In some of the case it is closed under ...
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26 views

Importance of universal enveloping algebras and Poincare-Birkhoff-Witt theorem.

Let $L$ denote a Lie algebra and $U(L)$ denote its universal enveloping algebra. I am trying to see why universal enveloping algebra and PBW theorem are important. Precisely: Given any ...
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0answers
100 views

A differential equation to determine a tangent vector

Let $c_1,c_2,c_3\in\mathbb{R}$. Suppose $a,b,c,d$ are smooth curves from $(-1,1)$ to $\mathbb{R}$ such that $a(0)=d(0)=1$ and ...
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0answers
49 views

The cohomology of $\mathrm{GL}_n$ over an algebraically closed field

How does one go about computing the cohomology groups $H^*(\mathrm{GL}_{\kern{0.1em}{m}}(\overline{\mathbb{F}}_p),M)$? I am particularly interested in the case when $M$ is an algebraic representation. ...
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The product of dg Lie algebras

I am trying to understand what are products and coproducts in the category of dg Lie algebras. I am okay with coproducts. For products, however, this Wikipedia article says that given ...
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2answers
34 views

A semisimple Lie group has no character; Am I right?

Let $G$ be a compact connected Lie group with semisimple Lie algebra ${\frak g}$. With the following reasoning, I show that there is no non-trivial Lie group homomorphism $$\chi:G\to S^1.$$ Is that ...
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36 views

Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
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19 views

Exponentiating an ``affine subalgebra''

Consider the Lie algebra $u(N)$. If I exponentiate an $x \in u(N)$, I will obtain an element of the $U(N)$ group. My understanding is that $exp(u(N))= \{exp(x)| x \in u(N)\}$ is, in fact, the group ...
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1answer
27 views

Algebraic groups and restricted Lie algebras

If $G$ is an algebraic group with coordinate algebra $A=\mathcal O(G)$, say over a field $k$ of characteristic $p$, then its Lie algebra $\mathfrak g$ can be endowed with the structure of a restricted ...
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23 views

Topologies of partially exponentiated lie algebras, especially in regard to $SU(2)$

Consider the fundamental respresentation of $\mathfrak{su}(2)$ given in terms of the Pauli matrices as $\mathfrak{su}(2) = \langle ...
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22 views

Is $V \otimes V$ a $g \otimes g$-module?

Let $g$ be a Lie algebra and $V$ a $g$ module. Then $V \otimes V$ is a $g$ module under the action $X.(x \otimes y) = X.x \otimes y+x \otimes X.y$, $x, y \in V$, $X \in \mathfrak{g}$. Is $V \otimes V$ ...
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1answer
18 views

What's its use of the nonsingular 2-step nilpotent Lie algebras

What's its use of the nonsingular 2-step nilpotent Lie algebras which form an a class of 2-step nilpotent Lie algebras ? Recall: A $2$-step nilpotent Lie algebra $N$ is non-singular if $ad X : N \to ...
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6answers
715 views

Getting started with Lie Groups

I am looking for some material (e.g. references, books, notes) to get started with Lie Groups and Lie Algebra. My motivation is that I (eventually) want to understand the theory underpinning papers ...
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how to show that the representation of $SL(2, \mathbb{C})$ is holomorphic

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to ...
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Problems with Matrix of irreducible representation of SO(3)

I'm working out the irreducible representation of $SO(3)$. Let's call $R_{\theta}$ as a general rotation and $Y_{m}^{l}\left(\eta,\varphi\right)$ the spherical harmonics. I now like to have the matrix ...
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1answer
31 views

Nilpotent Lie algebra

Help! I cannot solve this exercise If $g$ is a nilpotent Lie algebra, the two following assertions are equivalent: a) $\{x_1,\ldots,x_k\}$ is a minimal system of generators; b) $\{x_1 + g',\ldots, ...
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1answer
17 views

Lie algebra is nilpotent iff all two dimensional subalgebras are abelian?

I'm trying to prove that if $\mathfrak{g}$ is a Lie algebra over an algebraically closed field and every 2-d subalgebra is abelian then $\mathfrak{g}$ is nilpotent. By an induction all I need to show ...
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1answer
25 views

Classification Problems in Lie algebra

My field of interest is Lie group analysis of PDEs, currently I am struggling with techniques for classification of Lie algebra into mutually conjugate classes of 1- and 2-dimensional sub-algebras ...
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0answers
14 views

Centralizer of Maximal Toral Subalgebra of $L$

I am trying to understand the proof of the following result from Humphreys Lie Algebra book: Let $L$ be a semisimple complex Lie Algebra.Let $H$ be Maximal toral subalgebra of $H$,Let $C_L(H)$ ...
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18 views

Abstract Lie algebra

For six dimensional Lie algebra with non-zero Lie brackets defined as follow: $[e_{1}, e_{3}] = -e_{1}, [e_{1}, e_{6}] = -e_{2}, [e_{2}, e_{3}] = -e_{2}, [e_{2}, e_{4}] = e_{1}, [e_{2}, e_{5}] = ...
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1answer
12 views

Every Lie algebra contains a maximal proper Lie subalgebra

I am working though the proof of Proposition 6.2 in Erdmann's "Introduction to Lie Algebras". I can't verify that every Lie subalgebra of $L \subseteq \mathfrak{gl}(V)$ contains a maximal (proper) ...
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1answer
23 views

Compute the associated induced Lie algebra action $\text{d}\pi$

Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
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1answer
13 views

Let $F$ be a field and let $L = b(n,F)$ be the Lie algebra of $n×n$ upper triangular matrices and $V = {F^n}$ .

$\space$ Let $F$ be a field and let $L = b(n,F)$ be the Lie algebra of $n×n$ upper triangular matrices and $V = {F^n}$ . Let $e_{1} , ... , e_{n}$ be the standard basis of $F^n$. For $1 \le r \le n$ , ...