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)

2
votes
1answer
13 views

Identification of $H$ with $H^{*}$ relativ the Killing-form

Let $H$ be a maximal toral subalgebra of a semisimple Lie Algebra $L$. The identification of $H^{*}$ and H relativ the Killing-form says, that to $\phi\in H^{*}$ corresponds the unique element ...
1
vote
1answer
38 views

Show Lie bracket left invariant

I want to prove from the definition that the Lie bracket $[X,Y]$ of two left-invariant vector fields $X,Y: G \rightarrow TG$ where $G$ is a Lie group is again left-invariant. Left-invariance ...
1
vote
0answers
17 views

Choosing $(\alpha, \alpha)$ for root system $A_{1}$

I am looking at the Lie Algebra $L=\mathfrak{sl}(2, \mathbb{F})$ withthe root system $A_{1}=\{-\alpha, \alpha\}$. Let $\mathfrak{U}(L)_{\mathbb{Z}}$ the lattice in $\mathfrak{U}(L)$ with ...
2
votes
1answer
28 views

Lie algebra and Lie group cohomology, reference request

Who can give me a good reference (better if introductory/motivated) about Lie group cohomology, Lie algebra cohomology, and the link between the two? Thanks.
0
votes
1answer
34 views

equivalence of Lie group and Lie algebra intertwiner

I encountered this problem while working on my research. Let $G$ be a Lie group, and consider an intertwiner of the complex representations (possibly infinite-dimensional) $$ \pi:G\rightarrow ...
0
votes
1answer
18 views

How to find maximal dimension abelian subalgebra in finite Lie Algebra?

Is there any well known algorithm how to find maximal dimension abelian subalgebra in finite dimension Lie Algebra? If there is a built-in routine in some computer algebra system, it is the most ...
2
votes
1answer
27 views

Commutator ideal of reductive Lie algebra

I'm working through Fulton and Harris's book on Representation theory, and I've just done the exercise where I had to show: If $\mathfrak{g}$ is a reductive Lie algebra (defined as $Z(\mathfrak{g}) = ...
0
votes
1answer
32 views

Regarding proof of Theorem 3.3 in Humphreys (Lie alg. and Rep.)

Theorem 3.3. of Humphreys goes something like this: given a subalgebra $\mathfrak{g}$ of $\mathfrak{gl}(V)$ where $V$ is nonzero, finite-dimensional and $\mathfrak g$ consists of nilpotent ...
2
votes
1answer
30 views

References and suggestions about the elementary theory of Lie groups and Lie algebras

I am looking for suggestions on how to approach the field of Lie groups and Lie algebras. I am acquainted with both the elementary algebraic concepts, having studied from Bourbaki's "Algebra I-III", ...
2
votes
1answer
31 views

The deconposition of $\mathfrak{so}(V \oplus V^*)$

Let $V$ be an n dimensional real vector space and $V^*$ be the dual vector space. We have a non degenerate inner product $(\centerdot,\centerdot)$ in $V\oplus V^*$ such that $(v+\xi , ...
2
votes
0answers
20 views

Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: ...
1
vote
0answers
28 views

$Z(\mathcal{L}^{(n)}) \subset Z(\mathcal{L})$ for solvable Lie algebras?

$X$ Banach space. $\mathcal{L} \in B(X) $ is solvable Lie Algebra. Then for some n, $\mathcal{L} \supset \mathcal{L}^{(1)}=[\mathcal{L},\mathcal{L}] \supset ...
1
vote
1answer
19 views

Every base of a root system arises as indecomposable positive roots of a regular element?

I'm confused about a line in the Theorem p48 in Humphrey's's book on Lie Algebras. He's proving that every base $\Delta$ of a root system $\Phi$ arises as the set of $\Delta(\gamma)$ of ...
1
vote
1answer
16 views

Adjoint map is Lie homomorphism

The Jacobi identity of a Lie algebra says that $ad: \mathfrak g \to End(\mathfrak g)$ is a derivation. I am a bit emberassed but what is the easieast way to see that for every $X \in \mathfrak g$, ...
2
votes
0answers
24 views

signature function of Weyl group element in LieArt

I am currently using LieArt Mathematica package for some calculations in Lie algebra, I am wondering if there is a way to know what is the signature of a Weyl group element, it seems the package can ...
1
vote
0answers
18 views

How do I know the length of a Weyl group element from its Weyl orbit result

How do I know the length of a Weyl group element from its Weyl orbit result? For example, I know that $[2,2]$ under $s_1s_2s_1$ transforms into $[-2,-2]$,but given the result, how can I tell ...
1
vote
1answer
19 views

Finding the tangent space of $Z(U(n))$

Recall that the center of the matrix unitary group for a given $n\in \mathbb N$ is\begin{equation*}Z(U(n))=\{\omega I:|\omega|=1\}\end{equation*} I'm trying to find the tangent space at the identity ...
0
votes
0answers
29 views

What are the generators of the coset of SU(2)/U(1)?

I need to understand the coset, $SU(2)/U(1)$ for the fundamental representation. How would I go about doing so? From my understanding of coset it means that any transformation of $SU(2)$ mod a $U(1) ...
1
vote
1answer
38 views

Coefficients of positive roots in term of simple roots

Let $\Phi$ be an irreducible root system and $\Phi^+$ be positive root system and $\Delta$ be base. For every positive root $\beta=\sum_{\alpha \in \Delta}m_\alpha\alpha$, the numbers $m_\alpha$ are ...
3
votes
1answer
69 views

Symplectic group action

Let $(M,\omega)$ be a symplectic manifold. We say that a group action $\phi: G \times M \rightarrow M$ is symplectic if each $\phi(g,.)$ is a symplectomorphism. Now, I am going through some lecture ...
1
vote
1answer
15 views

Lie algebra associated to Lie group of algebra automorphisms

I'm working through Fulton and Harris's Representation Theory, and I'm stuck on Exercise 8.27. I'm trying to show that if $A$ is an algebra and $G$ is the Lie group of algebra automorphisms ...
1
vote
3answers
48 views

Equality on pg. 40 of Humphrey's Lie Algebras, $\kappa(t_\lambda,t_\mu)=\sum_{\alpha\in\Phi}\alpha(t_\lambda)\alpha(t_\mu)$?

I don't understand part of an equality on page 40 of Humphrey's book on Lie Algebras. Suppose $L$ is a semi-simple Lie algebra over an algebraically closed field of characteristic $0$, and $H$ a ...
2
votes
0answers
44 views

Why are chains topologically analoguous to distributions?

This question is related to my other question here but is different enough that I thought I might ask separately. At the nLab page on rational homotopy theory it is stated that chains are ...
2
votes
0answers
30 views

What is the connection between $\widehat{\mathbb Q G}$ and distributions near the identity of $G$?

I'm studying Quillen's rational homotopy theory and trying to understand this MathOverflow description of Quillen's functor provided by Hiro Lee Tanaka. When discussing connections between how ...
1
vote
1answer
42 views

Steinberg´s formula for $A_{1}$

I am trying to show, that Steinberg´s formula on the case $A_{1}$ yields the same result as the Clebsh-Gordan-formula, but I get a quite confusing result. Let $V(\lambda)$, $V(\lambda´)$ and ...
3
votes
1answer
118 views

Finalising proof from Humphreys´ “Introduction to Lie Algebras and Representation Theyory”

$L=\mathfrak{sl}(2, \mathbb{F})$ with standard Chevalley basis $(x, \ y, \ h)$ and $a, \ c\in \mathbb{Z}^{+}$. Humphrey gives a Lemma in chapter 26.2 saying: ...
1
vote
0answers
39 views

Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
2
votes
0answers
17 views

Real version of Harish-Chandra-Itzykson-Zuber integral

I'm interested in an integral of the form $$ \int_{O(d)} \exp\left(-\frac{1}{2}\mathrm{trace}(CUAU^T)\right)dU $$ where the integration is with respect to the Haar measure on the orthogonal group, ...
3
votes
1answer
32 views

Why do roots span dual space of maximal toral subalgebra?

Suppose $\Phi$ is the root system of a semi simple Lie algebra with maximal toral subalgebra $H$. I read that $\Phi$ spans $H^\ast$. The Killing form on $H$ is nondegenerate, so $H\cong H^\ast$ by ...
1
vote
0answers
28 views

Computing the Cohomology of Lie groups

In Bredons "Topology and Geometry" [Chapter V, section 12] the following theorem is proven: If $G$ is a compact connected Lie group its $k$-th cohomology $H^k(G,\mathbb{R})$ is isomorphic to the ...
1
vote
1answer
27 views

Adding tori to semi-simple groups

Let $G$ be a complex, connected, semi-simple Lie group (throw in simply connected if you like) with Lie algebra $\mathfrak g$. Let $T \subseteq B$ be a maximal torus and choice of Borel, respectively. ...
3
votes
1answer
21 views

How to embed $U(1)$ (or other groups) into a bigger group, using Dynkin diagrams

I am trying to find the embedding and the branching rules for some group decompositions. For example, I consider $E_7$ and its maximally compact subgroup $SU(8)$ and I want to "see" how the Dynkin ...
3
votes
2answers
38 views

$\mathfrak g = [\mathfrak g,e]\oplus {\rm Ker}({\rm ad}f)$ for an $\mathfrak {sl}_2$-triple $\{e,h,f\}$.

Let $\mathfrak g$ be a finite dimensional semisimple lie algebra over $\mathbb C$. Let $\{e,h,f\}$ be an $\mathfrak{sl}_2$-triple in $\mathfrak g$ (i. e. with relations $[h,e] = 2e$, $[h,f]=-2f$ and ...
5
votes
1answer
38 views

Formula for $\theta:\mathfrak{P}(L)^{G}\to \mathfrak{P}(H)^{W}$ for $\mathfrak{sl}_2$; exercise in Humphrey

Let $L=\mathfrak{sl}(2,\mathbb{F})$ with standard basis $(x, y, h)$ and dual basis $(x^{*}, y^{*}, h^{*})$, $H$ a CSA, $W$ the Weyl group and $G=\operatorname{Int}L$. Let $\mathfrak{P}(L)^{G}$ be ...
1
vote
1answer
26 views

The fundamental vector fields of a principal bundle are vertical.

Let $p:P\to M$ be a principal $G$-bundle. To each $A$ in the Lie algebra of $G$ corresponds a fundamental vector field $A^*$ on $M$ defined by $$A^*_u=\frac{d}{dt}|_{t=0} u(exp(tA))$$ How can we see ...
2
votes
1answer
46 views

Left and right action?!

The adjoint of the adjoint representation $Ad^* : G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*, (g,x) \mapsto Ad^*_{g}(x)$ is a group action on the dual space of the Lie algebra. Now, we said ...
1
vote
0answers
22 views

Lie algebra operations from lie group

According to wikipedia, if $G$ is a closed subgroup of $GL(n, \mathbb{R})$ then the Lie algebra of $G$ can be thought of informally as the matrices $m$ of $M(n, \mathbb{R})$ such that $1 + εm$ is in ...
0
votes
1answer
8 views

Compute variation left action subgroup

I consider a Lie group $G$, with a group element $g$ parametrised in some manner with parameter $\theta_i$, $i=1,\cdots, \dim G$. Suppose that $K\subset G$. I want to compute the variation of an group ...
2
votes
1answer
23 views

Criterion for semisimplicity for $\mathfrak{so}_6(\Bbb C)$

I'm trying to prove that $\mathfrak{so}_6(\Bbb C)$ is semisimple. There exists a criterion which says that, given a Lie algebra $L\le\mathfrak{gl}(V)$, where $V$ is an irreducible $L$-module, then ...
2
votes
0answers
48 views

Standard cyclic module of sl2

Let $L=\mathfrak{sl}(2, \mathbb{F})$, $B$ a standard Borel subalgebra. I am trying to solve exercise 20.4 from J.E. Humphreys "Introduction to Lie Algebras and Representation Theory", but I am stuck. ...
4
votes
0answers
39 views

Ideals of Lie-algebras

I am wondering whether the following claim is true: Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra and $V$ some vector subspace of $\mathfrak{g}$. Claim: $V$ is an ideal of $\mathfrak{g}$ ...
0
votes
0answers
15 views

Are the elements of the adjoint represetnation normal operators

Given a Lie group $G$ with Lie algebra $\mathfrak{g}$ on has the adjoint action of each $g\in G$ given by $Ad_g(\mathfrak{g})$. Is $Ad_g: \mathfrak{g} \rightarrow \mathfrak{g}$ a normal operator ...
1
vote
0answers
12 views

The algebra of $W$-invariant polynomial funktions sl2

Let $L=\mathfrak{sl}(2,\mathbb{F})$, $H$ a borel subalgebra, $\Delta=\{\alpha\}$ a base of the corresponding root system and $W$ the Weyl group. Let $\lambda=\frac{1}{2}\alpha$ be the fundamental ...
0
votes
1answer
98 views

Equivalent definitions of positive root system

I begin with a definition of positive root systems of a root system over Euclidean space. A subset $\Delta$ of root system $\Phi$ is called a simple root system (or base) in $\Phi$ if (1) $\Delta$ ...
4
votes
2answers
94 views

Lie Groups/Lie Algebra - Applications?

I studied Lie Groups and Lie Algebras as part of my Masters back in the 1970s. Although very elegant and beautiful, it seemed to its own little world, I never saw the connection with other branches of ...
0
votes
1answer
50 views

Given adjoint action find original matrix.

Given the Adjoint action of a matrix; $\text{Ad}(g) X_1 = g \, X_1 \, g^{-1} = X_2 $. Where g is in a (matrix) Lie group, $X_1,\; X_2$ are from the Lie algebra, can a $g$ be written in terms of the ...
0
votes
0answers
30 views

Nonlinear Lie group from Fulton & Harris

On page 138 of my copy of the celebrated Representation Theory by Fulton & Harris, a proof is outlined to show that the real group of $3\times 3$ upper-triangular unipotent matrices modulo a ...
0
votes
0answers
16 views

Stable Cartan under two involutions

Let $\mathfrak{g}$ a complex semisimple Lie algebra, and let $\theta$ be an involutive automorphism of $\mathfrak{g}$. The following fact is known. Any parabolic subalgebra contains a ...
7
votes
0answers
146 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
1
vote
0answers
31 views

Universal enveloping algebra of sl2

I am currently trying to proof, that $x-1$ is not invertible in the universal enveloping algebra $\mathfrak{U}(\mathfrak{sl}(2,\mathbb{F}))$ of $\mathfrak{sl}(2, \mathbb{F})$, but I still struggle ...