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

learn more… | top users | synonyms (1)

0
votes
0answers
16 views

Does the trivial character always show up as a weight?

Let $G$ be a linear algebraic group, $T$ a subtorus of $G$ of dimension $\geq 1$. Let $\mathfrak g$ be the Lie algebra of $G$. Then the Ad operator $$\textrm{Ad } : G \rightarrow \textrm{GL}(\...
2
votes
1answer
45 views

Connecting the regular representation of $\mathfrak{so}(3)$ and the exterior algebra of $\mathbb{R}^3$

It is well known that the regular representation of $\mathfrak{so}(3)$ is the so-called "cross product" matrix $A(x)$ which follows $A(x)y = x\times y$, and $x,y\in\mathbb{R}^3$, while the cross ...
0
votes
0answers
18 views

augmentation ideal for universal enveloping algebras

Let $L$ be a restricted Lie algebra with the restricted enveloping algebra $u(L)$ over a field $F$. Let $ω(L)$ denote the augmentation ideal of $u(L)$ which is the kernel of the augmentation map $\...
1
vote
1answer
27 views

Existence of Certain Lie Groups

Let $\mathfrak{h}$ be a Lie algebra (not necessarily finite dimensional). Does there necessarily exist a Lie group $G$ such that for the Lie algebra corresponding to $G$, denoted $\mathfrak{g}$, we ...
0
votes
1answer
48 views

Ordering on the weight lattice

When given a finite dimensional complex Lie algebra $\mathfrak{g}$ that is also semisimple and a choice of Cartan subalgebra $\mathfrak{h}$ we may talk about its weight lattice $\Lambda_{W} $ in $\...
0
votes
1answer
40 views

Meaning of generators in Lie Algebra of PDE

Consider some PDE involving a scalar function $u(x,t)$ with two independent variables $x$ and $t$. Assume that this PDE has a Lie Algebra spanned by the following generators, $X_1=\partial_x,\quad ...
0
votes
1answer
16 views

Lowering a non-zero weight vector gives a non-zero vector (representation of $\mathfrak{sl}(2)$)

In Lie algebras we study $\mathfrak{sl}(2)$ (the complex span of the usual matrices $X,Y,H$ where $X$ and $Y$ are the raising and lowering operators respectively). The defining commutator relations ...
2
votes
2answers
25 views

Soft question about Lie Groups and 3D rotation

Let $R(\phi, \boldsymbol{n})$ be a member of Lie Group SO(3). According to Wikipedia If $R(\phi, \boldsymbol{n})$ denotes a counter-clockwise 3D rotation through an angle $\phi$ about the axis ...
2
votes
1answer
38 views

Showing $[L,rad(L)]$ is nilpotent.

Suppose L is a finite dimensional Lie Algebra over an algebraically closed field of characteristic zero. I want to show that $[L,rad(L)]$ is nilpotent. I was given a hint that all operators of the ...
0
votes
1answer
17 views

Proof of Jordan Decomposition of derivation in Lie Algebras, 'Acts Diagonalisably'

My understanding is $[x,y] \in L_{\lambda + \mu} \Rightarrow (\delta - (\lambda +\mu )I_L)^m [x,y]=0$ Somehow, the $m$ has changed into a $1$ and I believe that is because of the 'acts ...
5
votes
1answer
63 views

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 $\mathbb{R}$ $\mathbb{g}=\begin{...
0
votes
0answers
16 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 \mathfrak{...
0
votes
1answer
34 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 ...
5
votes
1answer
202 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 ...
3
votes
1answer
61 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 ...
2
votes
2answers
37 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 $...
0
votes
2answers
32 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 groups?...
0
votes
0answers
15 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 ...
0
votes
0answers
33 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 ...
1
vote
1answer
32 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 $$ (*) \ \ \ \ \ \ \ ...
3
votes
1answer
69 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 = \{\...
2
votes
1answer
30 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 ...
2
votes
1answer
27 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 $V=\...
0
votes
0answers
51 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 ...
0
votes
1answer
19 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 $\...
0
votes
1answer
16 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 $A(...
0
votes
1answer
55 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 ...
2
votes
0answers
62 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. ...
0
votes
0answers
34 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 ...
3
votes
0answers
34 views

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 $\mathfrak{g},\...
1
vote
2answers
38 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 ...
1
vote
0answers
20 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 $U(...
1
vote
1answer
29 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 ...
0
votes
0answers
29 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 \frac{i\sigma_1}{2},\frac{i\sigma_2}{2},\frac{i\sigma_2}{2}\rangle_{\...
1
vote
0answers
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 ...
0
votes
1answer
22 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 ...
0
votes
0answers
21 views

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 GL(...
2
votes
1answer
25 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} \mathfrak{e}\stackrel{\...
1
vote
0answers
20 views

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 ...
0
votes
1answer
33 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, ...
1
vote
1answer
31 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 ...
0
votes
0answers
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 $V \otimes V$ ...
0
votes
1answer
50 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 ...
0
votes
0answers
17 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 $C_L(H)$ ...
3
votes
1answer
92 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 ...
0
votes
0answers
20 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}] = e_{2},...
1
vote
0answers
29 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 $L$-...
0
votes
1answer
16 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) ...
0
votes
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 $1 \le r \le n$ , ...
0
votes
0answers
28 views

Relationship between Casimir and index of a representation of a Lie algebra.

In several QFT textbooks (namely, those of Peskin and Shroeder and of Schwartz) there is presented an identity for representations of Lie algebras, $$ d(R) C_2(R) = T(R) d(G),$$ where $d(R)$ is the ...