Questions tagged [lie-algebras]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
16 views

How does the lie exponential map act on tangent vectors?

I'm currently attempting to understand a little bit about how the exponential map works in general. I'll try to lay out what it is I think I've understood and where I think the problem lies. If I have ...
0
votes
0answers
12 views

Worked Examples of Root System Calculation

I am looking for some worked examples of questions like "Compute the root system of the special orthognal Lie algebra $\mathfrak{so}_{2n}$". I understand the theory but find the computations ...
1
vote
1answer
24 views

Derivate of matrix conjucation

Consider the map from $GL(R,2)$ which sends a matrix $X$ to the map ($Y\mapsto XYX^{-1}$). This is a map smooth map $f:GL(R,2)\rightarrow End(GL(R,2)$ so we should be able to calculate its derivative. ...
0
votes
0answers
21 views

Derivative of $Ad:SU(2)\rightarrow \text{Aut}(\text{Lie}(SU(2))$

In full generality $\text{ad}$ is the derivative of $\text{Ad}$.I would like to see this explicitly. We want to calculate $d_e\text{Ad}:T_eSU(2)\rightarrow T_{Ad(e)}O(3)$. If we want to calculate this ...
0
votes
1answer
23 views

Adjoint map $\text{Ad}:G\rightarrow \text{Aut}(\mathfrak{g})$ is smooth.

Let $G$ be a lie group and let $\mathfrak{g}$ its lie algebra. Its not clear to me that $\text{Ad}:G\rightarrow \text{Aut}(\mathfrak{g})$ is smooth. It is clear to me that if we have a matrix group ...
3
votes
0answers
24 views

Smooth lifting criteria of smooth vector fields given smooth surjective submersion whose fibers are connected.

I am working on an exercise: Suppose $F : M \to N$ is a smooth submersion, where $M$ and $N$ are positive-dimensional smooth manifolds. Given $X \in \mathfrak{X}(M)$ and $Y \in \mathfrak{X}(N)$, we ...
1
vote
0answers
17 views

Understanding and intuition for the adjoint action of a Lie group on its Lie algebra

We denote $G$ a Lie Group and $\mathfrak{g}$ its lie algebra. I would like to what the intuition behind these adjoint actions $\text{Ad}$ and $\text{ad}$ are. Could you please give a concrete example ...
1
vote
1answer
15 views

Each Weyl group orbit in the character lattice of $V$ contains exactly one dominant weight

Let $V = \mathbb{C}^3 \otimes \mathbb{C}^3$ be a representation of $G = SL_3(\mathbb{C})$. The weights of this representation is the set of $\varepsilon_i + \varepsilon_j$ for $i, j = 1, 2, 3$, where $...
0
votes
0answers
9 views

Unipotent upper triangular matrices with integer entries is Zariski dense

Let $N$ be the group of matrices $\begin{bmatrix} 1 & z \\ 0 & 1 \end{bmatrix}$ for $z \in \mathbb{C}$, let $\Gamma$ be the subgroup of $N$ with $z \in \mathbb{Z}$. I wish to show that $\Gamma$...
0
votes
0answers
72 views

Calculate of Lie Bracket of $R2×S1$

We have R2×S1 equipped multiplication $(x,y,z)$.$(x',y',z')=(x+x',y+y',zz'exp(ixy'))$ I know how to show that it is a Lie group and I think that its Lie algebra is $R^3$(no?) Then we have a base $(e1,...
0
votes
1answer
20 views

Understanding the textbook of Humphreys Lie algebras

I'm studying Humphreys Introduction to Lie Algebras and Representation Theory. I do not understand the second paragraph in page 26. Given a representation $\phi : L \to \mathfrak gl (V)$, the ...
0
votes
1answer
20 views

The lowest weight of a representation is $w_0 \lambda$

There is the well-known fact that if $\lambda$ is the highest weight of $V$ a finite-dimensional irreducible $\mathfrak{g}$-module, then $w_0 \lambda$ is the lowest weight (here $\mathfrak{g}$ is any ...
2
votes
0answers
21 views

Expression for the Lee form

On a Hermitian manifold $(M,J,g)$ with associated fundamental form $\omega(X,Y)=g(JX,Y)$ we have an $L^2$ inner product $\langle \cdot,\cdot \rangle$ on $p$-forms $\alpha,\beta$ given by integrating $\...
2
votes
1answer
28 views

$\Phi$-extreme weights and the Weyl group orbit of the highest weight

$\newcommand{\g}{\mathfrak{g}}$ Let $P(\g)$ be the weight lattice of $\g$ a semisimple Lie algebra over $\mathbb{C}$, and $P_{++}(\g)$ the set of dominant integral weights. A subset $\Psi \subset P(\g)...
1
vote
1answer
41 views

Highest-weight module over $\mathfrak{sl}_2$

I'm reading the following example, which I have trouble understanding: Let $\mathfrak{g}=\mathfrak{sl}_2$ and $\lambda=m\Lambda_1$. Consider the irreducible module $V$. Then the vectors in the ...
1
vote
0answers
36 views

Finding the set of weights for a representation of a Lie group

Let $V = \mathbb{C}^3 \otimes \mathbb{C}^3$ be a representation of $G = SL_3(\mathbb{C})$. I want to find the weights of this representation in terms of the functionals $\varepsilon_i$ for $i = 1, 2, ...
1
vote
1answer
19 views

Is the radical of a Lie algebra equal to the radical of its Killing form?

A Lie algebra $\mathfrak{g}$ is semi-simple if the maximal solvable ideal ${\rm rad}(\mathfrak{g})$ is trivial (let's take this as the definition for this question). Cartan's Second Criterion says ...
0
votes
0answers
21 views

The exponential map as a local diffeomorphism between Lie groups [closed]

It is known that the exponential map $ exp \colon M_n(\mathbb{C}) \to GL(n, \mathbb{C}) $ is a local diffeomorphism from a neighborhood of $0 \in M_n(\mathbb{C})$ to the identity $ I_n \in GL(n, \...
1
vote
0answers
28 views

Reference request: homomorphisms of $\widehat{\mathfrak{sl}_2(\mathbb C)}$-modules

For a Lie algebra $\mathfrak g$ one can construct $\hat{\mathfrak g}$-modules at level $k$ by taking induced representations from $\mathfrak g$-modules, and then quotienting by the maximal submodule. ...
0
votes
1answer
36 views
+50

Is there a known representation for this set derived from a Lie group $\subseteq \text{SU}(n)$?

Let $G \subseteq \text{SU}(n)$ be a Lie group for $n \in \mathbb{N}$, with a proper Lie subgroup $H \subset G$ having Lie algebra $\mathfrak{h}$ such that $H = e^{\mathfrak{h}}$. Let $g_0 \in G$ be an ...
0
votes
0answers
15 views

Ideals and derived algebras. [duplicate]

Consider L is an ideal in $g/ g '$ of codimension 1. Let $ \pi: g \longrightarrow g / g '$ homomorph be canonical. So, $ \pi^{-1} (L)$ um is ideal in g. In fact, if $x \in \pi^{-1}(L)$ and $\in g$, ...
2
votes
1answer
49 views

Is the Ricci tensor of a symmetric space proportional to the killing form?

In Besse's book on Einstein manifolds, one can read the following theorem 7.73 Theorem: The Ricci curvature of a Riemannian symmetric space satisfies: $$r=-\frac{1}{2}B_{\vert\mathfrak p}$$ Here $r$ ...
2
votes
0answers
37 views
+200

Let $a\in\mathfrak{n}^+$ st. $[a,f_i]=0$ for all $1\leq i\leq n$ then $a=0$ (lemma 1.5 in Kac's inf. dim. Lie algebras)

I'm trying to understand the proof of the following lemma: Let $a\in\mathfrak{n}^+$ st. $[a,f_i]=0$ for all $1\leq i\leq n$. Then $a=0$. We are in the setting of a Kac-Moody algebra $\mathfrak{g}=\...
0
votes
0answers
14 views

How look the Lie algebra cohomology complex(es) $C^n(L,M)$ look? [closed]

Let $k$ be a field, $L$ is the Lie algebra over a $k$-vector space $L$, and $M$ is the $L$-module. The main question is: How to show or verify, that $C^n(L,M) = \hom(\Lambda^n L,M)$? Why is setting $...
1
vote
1answer
21 views

Calculating the Lie bracket on the Heisenberg algebra of $H=Z\times S^1$

I'm working through Mechanics and Symmetry by Marsden and Ratiu. Let $(Z,\Omega)$ be a symplectic vector space and define on $H:=Z\times S^1$ the operation $$(u,\exp i\phi)(v,\exp i\psi)=(u+v,\exp i[\...
6
votes
2answers
56 views

How to prove the rep of SU(2) on homogeneous polynomials in 2 variables is irreducible?

The group $\mathrm{SU}(2)$ has a tautologous representation on the space $\mathbb{C}^2$ and thus a representation on the $d$th symmetric power $S^d (\mathbb{C}^2)$. What's the easiest way to prove ...
0
votes
2answers
38 views

Example for a faithful and finite dimensional representation over a sovable and finite dimensional Lie-Algebra

I am new to StackExchange. I am learning about Lie-Algebras and I was wondering whether somebody can give me an example for a finite dimensional and faithful representation of a sovable and finite ...
1
vote
0answers
34 views

The Heisenberg group/algebra and Symplectic Vector Spaces

I have some questions about the relationship between Heisenberg groups/algebras and symplectic vector spaces. This is my first time properly dealing with many of these topics, so please be patient if ...
1
vote
0answers
30 views

Fixed points of the adjoint representation for an algebraic group

Let $G$ be a reductive affine algebraic group over $k$, and consider the adjoint action $G \xrightarrow{\text{Ad}} \text{GL}(\frak{g})$, where $\frak{g}$$=\text{Lie}(G)$ is the Lie algebra of $G$. I ...
1
vote
0answers
40 views

Lie groups over fields of finite characteristic

Does anyone have any good references on Lie groups over fields of finite characteristic? I am trying to find something comprehensive that shows what fails and what succeeds in comparison to Lie theory ...
0
votes
0answers
16 views

dual Lie algebra, dual Lie group, and Langland dual group

Are the following concepts somehow related? dual Lie algebra dual Lie group Langland dual group (say of a Lie group) We can take examples, for su(N) Lie algebra and SU(N) Lie group; or so(N) Lie ...
0
votes
1answer
23 views

Solvable Lie algebras. derived algebras and ideals

Let L be a vector subspace codimension $1$ in $g/ g'$, where g is a Lie algebra solvable and $g'=[g,g]$ derived algebra. Have: $g/g'$ is abelian In fact, if $x, y \in g$ $$[x+g',y+g']=[x,y]+g'= g'=0+...
0
votes
0answers
26 views

Trace orthogonality of representations of Lie group generators

Typically, given a (simple) Lie group $G$ we choose a basis for the associated Lie algebra $\mathfrak{g}$ so that in the fundamental, or defining, representation, the basis is trace-orthogonal, $$ \...
1
vote
0answers
32 views

presentation of a generic algebra

In the book $\ulcorner$Reflection Groups and Coxeter Groups$\lrcorner$ written by J. E. Humphreys, in the beginning of chapter 7 $<$Hecke algebras and Kazhdan-Lusztig polynomials$>$, it defines ...
0
votes
1answer
27 views

Bilinear form respect to the representation $V/W$ of Lie algebra

I got a problem from the book An Introduction to Lie Groups and Lie Algebras written by Kirillov. The exercise 5.1 says $$\begin{array}{l} \text { (1) Let } V \text { be a representation of } \...
2
votes
0answers
20 views

Solving/Rewriting SDEs in Non-Matrix Lie Groups

I'm working on trying to solve a state estimation problem in a non-matrix Lie group. I have found some good resources for state estimation in certain matrix Lie groups. For instance, in this paper ...
0
votes
0answers
43 views

Why does this equality holds for any choice of basis in $\mathfrak{g}$

Let $G$ be a lie group with lie algebra $\mathfrak{g}$ and let $M$ be a $G$-manifold. We denote the space of smooth differential forms on $M$ by $A(M)$. Let $E^i$ be a basis of $\mathfrak{g}$ and let $...
1
vote
1answer
44 views

Hopf algebra structure on universal enveloping algebra?

Let $\mathfrak g$ be a Lie algebra. Show that on $U(\mathfrak g)$ (universal enveloping algebra) there is a natural Hopf algebra structure induced by the Hopf algebra structure on the tensor algebra $...
1
vote
0answers
39 views

Roots of the same length differ by a permutation by the Weyl group

I am to show that for $ G \subset GL(n,\mathbb{C})$ a classical group, $(\cdot, \cdot)$ the usual inner product on $\Phi$ the root system of $G$, and $(\alpha, \alpha) = (\beta, \beta)$, there exists $...
0
votes
0answers
6 views

PBW theorem for symmetric Leibniz algebras

A Leibniz algebra is called a symmetric Leibniz algebras if it is right and left Leibniz algebra at the same time. I need a reference in which the PBW theorem has been studied for symmetric Leibniz ...
3
votes
0answers
26 views

Lie transported commutator ?= commutator of Lie-transported vectors [Schutz]

I am working through Schutz's Geometrical methods in mathematical physics. Stuck in 3.8, on Forbenius' theorem. My question is about the very last step. Let there be an $n$-dimensional manifold $M$ ...
0
votes
0answers
24 views

Definitions of the quadratic Casimir of $SU(N)$

Particle physics, especially QCD, deals a lot with $SU(N)$ and therefore also with $\mathfrak{su}(N)$. In the QCD literature it is normal to define the (quadratic) Casimir element in a representation ...
0
votes
2answers
49 views

An example for finding an explicit definition of “structure maps” [closed]

By googling, I found only What is meant by a “structure map”?. But I could not understand that for my example as follows: Let $L$ be a Lie algebra over $K$, and view $K$ as a trivial $L$-module (that ...
1
vote
1answer
33 views

An identity for elementary automorphisms of a semisimple Lie algebra

$\newcommand{\ad}{\text{ad}}$ $\newcommand{\Ad}{\text{Ad}}$ Let $\mathfrak{g}$ be a semisimple Lie algebra and let $X \in \mathfrak{g}$ be nilpotent. Then $\ad X$ is nilpotent and $\exp \ad X$ acts on ...
1
vote
0answers
16 views

Lie superalgebra sl(m|n)

I know sl(m|n) is the A-series Lie superalgebra by Kac But in some literature, people also use the terminology su(m|n), which seems to be subclass of A-series. I am wondering how these two are related ...
2
votes
0answers
48 views

Automorphism Aut/Inn/Out of the unitary group $U(N)$

Given a group $G$, we denote the center Z$(G)$, we like to know the automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences: $$...
1
vote
1answer
36 views

Definition of coroots in a semisimple Lie algebra

For $\mathfrak{g}$ a semisimple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra, $\Phi$ a root system with respect to $\mathfrak{h}$, the coroot $\check{\alpha} $ associated to a root $\alpha \in \Phi$...
0
votes
1answer
18 views

Integrality property of the root system of a semisimple Lie algebra

For $\mathfrak{g}$ a semisimple Lie algebra, $\mathfrak{h}$ a choice of Cartan subalgebra, $\Phi$ the set of roots, there is the integrality property that if $\alpha$ is a root then $c \alpha \in \Phi$...
2
votes
1answer
47 views

Does a Lie algebra generate the entire identity component?

Suppose we have a Lie group $G$ with Lie algebra $\mathfrak{g}$. If $G$ is connected, is it true that $\exp\mathfrak{g} = G$? If $G$ is not connected, does $\exp\mathfrak{g} = G_0$ where $G_0$ is the ...
1
vote
1answer
80 views

If $[X,Y]=0$ then $X=0$

Let M be a manifold and $X \in \mathfrak{X}(M)$ show that if for any $Y \in \mathfrak{X}(M)$ , $[X,Y]=0$ then $X=0$. I think we can use following theorem Theorem: $[X, Y]=0$ iff the flows of $X$ and $...

1
2 3 4 5
106