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.
5,278
questions
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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
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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 ...
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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.
...
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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 ...
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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
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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 ...
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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 ...
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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 $...
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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 $\Gamma$...
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72 views
Calculate of Lie Bracket of $R2×S1$
We have R2ĆS1 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,...
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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 ...
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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
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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
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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)...
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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 ...
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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 ...
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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, \...
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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.
...
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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 ...
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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
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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
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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}=\...
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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
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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 $$(u,\exp i\phi)(v,\exp i\psi)=(u+v,\exp i[\...
6
votes
2answers
56 views
How to prove the rep of SU(2) on homogeneous polynomials in 2 variables is irreducible?
The group $\mathrm{SU}(2)$ has a tautologous representation on the space $\mathbb{C}^2$ and thus a representation on the $d$th symmetric power $S^d (\mathbb{C}^2)$. What's the easiest way to prove ...
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2answers
38 views
Example for a faithful and finite dimensional representation over a sovable and finite dimensional Lie-Algebra
I am new to StackExchange. I am learning about Lie-Algebras and I was wondering whether somebody can give me an example for a finite dimensional and faithful representation of a sovable and finite ...
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0answers
34 views
The Heisenberg group/algebra and Symplectic Vector Spaces
I have some questions about the relationship between Heisenberg groups/algebras and symplectic vector spaces. This is my first time properly dealing with many of these topics, so please be patient if ...
1
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0answers
30 views
Fixed points of the adjoint representation for an algebraic group
Let $G$ be a reductive affine algebraic group over $k$, and consider the adjoint action $G \xrightarrow{\text{Ad}} \text{GL}(\frak{g})$, where $\frak{g}$$=\text{Lie}(G)$ is the Lie algebra of $G$. I ...
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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 ...
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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 ...
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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 $x, y \in g$
$$[x+g',y+g']=[x,y]+g'= g'=0+...
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26 views
Trace orthogonality of representations of Lie group generators
Typically, given a (simple) Lie group $G$ we choose a basis for the associated Lie algebra $\mathfrak{g}$ so that in the fundamental, or defining, representation, the basis is trace-orthogonal,
$$
\...
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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 ...
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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
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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 ...
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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
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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 $...
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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 $...
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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
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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$ ...
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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 ...
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2answers
49 views
An example for finding an explicit definition of “structure maps” [closed]
By googling, I found only What is meant by a āstructure mapā?. 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
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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 ...
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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
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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
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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
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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
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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 $...
