All Questions
2,633 questions
10
votes
2
answers
3k
views
References on Lie groups and dynamical systems
I'm interested in Lie theory and its connections to dynamical systems theory. I am starting my studies and would like references to articles on the subject.
- 18.7k
8
votes
0
answers
229
views
What can we say about the homogeneous spaces $E_8/E_7$ and $E_7/E_6$?
For the three exceptional compact Lie groups $E_8, E_7, E_6$ we have the inclusions
$$
E_6 \subseteq E_7 \subseteq E_8.
$$
What can we say about the the homogeneous spaces
$$
E_8/E_7, ~~~~ E_7/E_6?
$$
...
- 305
1
vote
0
answers
69
views
In a contact Lie algebra, when is the Reeb vector a semisimple element?
Below is a question I have come across in my research, and it seems like a question that has been answered (or at least asked) in the past; however, I have been unable to find any references that ...
- 63.9k
6
votes
2
answers
315
views
Is the triple product in a Freudenthal triple system fully symmetric?
I'm trying to learn about Freudenthal triple systems. Here is the definition given by Helenius [1], start of Section 5:
A Freudenthal triple system is a finite-dimensional vector space $V$
over a ...
- 2,493
7
votes
0
answers
125
views
Softwares to determine semi-simple types of Lie algebras generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of matrices
I wish to determine the type of a Lie algebra generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of square matrices with irrational elements. For example,
\begin{align}
n^+ =
\begin{pmatrix}
...
- 63.9k
2
votes
0
answers
146
views
Koszul differential of the complex $\bigwedge \mathfrak{g}^*$
Let $\mathfrak{g}$ be a finite dimensional Lie algebra. The definition of the Koszul differential is given in the article by Kumar and Vergne on equivariant cohomology page 133 as follow:
Let us now ...
- 63.9k
4
votes
0
answers
97
views
Why is the $A$-series root system best written in a vector space of one dimension higher?
In the classification of root systems, we have four families $A_n,B_n,C_n$, and $D_n$, and six exceptionals $E_6,E_7,E_8, F_4$, and $G_2$. For every non-exceptional case except $A_n$, the root system ...
- 123
6
votes
1
answer
256
views
A weight generalization of root systems?
For any simple complex Lie algebra $\frak{g}$, with a given choice of Cartan subalgebra $\frak{h}$, we have an associated root system $R \subseteq \frak{h}^*$. The properties of $R$ can be formalized ...
- 24.2k
2
votes
1
answer
168
views
Irreducible $G$-representations with unital algebra structure
Let us work over $\mathbb C$. Suppose that $G$ is a semisimple algebraic group and let $H \subset G$ be a maximal torus. Consider a dominant weight $\omega$, then one can associate a unique ...
- 63.9k
2
votes
1
answer
178
views
Prove: Lie algebra generated by two $n\times n$ shift matrices is $\mathfrak{so}(n,\mathbb{C})$ ($n$ odd) or $\mathfrak{sp}(n,\mathbb{C})$ ($n$ even)
I wish to have a proof for the following result:
Let $U_n$ be an $n\times n$ upper shift matrix, and $L_n = U_n^T$ be a lower shift matrix. For example,
$$
U_5 = \begin{pmatrix}
0 & 1 & 0 &...
- 1,336
7
votes
1
answer
317
views
Semisimple super Lie algebras
There is a classification of simple Lie algebras in $\text{Vec}_{\mathbb{C}}$ given by Dynkin diagrams. We then have 4 families of simple lie algebras, plus some exceptional ones.
Question: How about ...
- 7,746
19
votes
1
answer
2k
views
The Infinitesimal topos in positive characteristic
This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie ...
- 19.4k
2
votes
1
answer
229
views
Action of the negative Cartan-Weyl generators on a highest weight element
Let $\frak{g}$ be a complex simple Lie algebra of rank $l$. For $\frak{h}$ a choice of Cartan subalgebra, let $\alpha_1, \cdots, \alpha_r$ be the corresponding choice of simple roots, $X_{\alpha_i}, ...
- 123
3
votes
0
answers
101
views
Character formula for real representations
For an irreducible representation of a complex semisimple Lie algebra the Weyl character formula is well known. The real representations of a real semisimple Lie algebra are classified using their ...
- 103
2
votes
0
answers
55
views
Function annihilated by ideal in universal envelope
Let $G$ be a Lie group, $U(G)$ its universal enveloping algebra over $\mathbb C$ and let $J\ne U(G)$ be a left ideal. We consider $U(G)$ as the algebra of left-invariant differential operators on $G$.
...
CommunityBot
- 1
4
votes
1
answer
277
views
Jordan decomposition on the dual Lie algebra
$\newcommand\fg{\mathfrak g}\newcommand\gl{\mathfrak{gl}}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field, and let $G$ be a smooth, affine algebraic ...
CommunityBot
- 1
10
votes
1
answer
636
views
Killing form: clean proof that two spaces are orthogonal?
Let $G$ be a linear algebraic group whose Lie algebra $\mathfrak{g}$ is semisimple. Let $x$ be a regular semisimple element of $G$. Write $\mathfrak{t}$ for the Lie algebra of the maximal torus $T = C(...
- 954
3
votes
0
answers
202
views
The group of fixed points of an involution of a Weyl group
Let $R$ be a reduced root system in a vector space $V$ over $\mathbb Q$.
Let $W=W(R)$ denote its Weyl group.
Let $S\subset R$ be a basis of $R$ (a system of simple roots).
Let $D=D(R,S)$ denote the ...
- 14.2k
6
votes
1
answer
174
views
CE(g) for g infinite dimensional
On the nlab page for Chevalley–Eilenberg algebras, it defines $\operatorname{CE}(\mathfrak g)$ for $\mathfrak g$ finite dimensional, and then says "This has a more or less evident generalization ...
- 35.5k
-1
votes
1
answer
558
views
Representation of Lie algebra $\operatorname{SE}(2)$
When I read the paper Universal approximations of invariant maps by neural networks of Dmitry Yarotsky, it happens on page 36 that he used some concepts about the representation of Lie algebra of the ...
- 129
3
votes
0
answers
137
views
Lie algebra action from the action of $\mathfrak{sl}_2$-triple operators
Let $\mathfrak{g}$ be a complex simple Lie algebra and let $\{e,f,h\}$ be a $\mathfrak{sl}_2$-triple in $\mathfrak{g}$, i. e. we have commutation relations:
$$
[e,f]=h,~[h,e]=2e,~[h,f]=-2f.
$$
In ...
- 379
10
votes
1
answer
757
views
Can the numerator in Weyl's character formula be written as a determinant?
I paraphrase part of the wikipedia article on the Weyl character formula: Weyl character formula.
If $\pi$ is an irreducible finite-dimensional representation of a complex semisimple Lie algebra $\...
- 24.2k
3
votes
0
answers
111
views
Irreducible dimensions generating function for Lie algebra $\mathfrak{sl}_n$
Let $\lambda = \sum_{i = 1}^{n - 1} m_i \omega_i$ be the highest weight of irreducible representation $V(\lambda)$ of Lie algebra $\mathfrak{sl}_n$. As we know from the Weyl formula,
$$\dim V(\lambda) ...
- 645
8
votes
0
answers
688
views
An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich
Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$.
In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map
$$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
- 9,650
9
votes
2
answers
1k
views
Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$?
My questions may turn out to be related to Schur functors.
If $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda$ is the highest weight of an irreducible representation $V$ of $\mathfrak{...
- 9,650
7
votes
0
answers
171
views
$\mathfrak{sl}_2(\mathbb{Z})$'s properties as a Lie algebra over a ring
I was wondering what was known about $\mathfrak{sl}_2(\mathbb{Z})$ as a Lie algebra; in particular, what is known about its representation theory? I know of some texts which treat Lie algebras ...
- 820
4
votes
0
answers
189
views
If the Frobenius endomorphism of a characteristic $p$ ring is epimorphic, is it surjective?
MO question 19282 is about integral epimorphisms of commutative rings, and a counterexample is given to surjectivity. What about the case of the Frobenius endomorphism of a commutative, characteristic ...
- 4,492
3
votes
3
answers
264
views
Nontrivial Poisson relations for affine Poisson algebras
Let $A$ be a polynomial algebra over a field of characteristic $0$ in the variables $x_1,\dots,x_n$. Consider polynomials $f_1,\dots,f_m\in A$ and let $I$ be the ideal they generate in $A$. Moreover, ...
- 359
6
votes
0
answers
355
views
Homotopy transfer of cyclic L-infinity algebras
Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a ...
- 35.5k
1
vote
0
answers
185
views
Exact sequence of L-infinity-algebras
We call a sequence of $L_\infty$-algebras (weak) maps
$$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$
is exact if it is exact on the the underlying chain complexes level.
Thought I don't know ...
- 35.5k
12
votes
2
answers
855
views
Groups associated with infinite dimensional Lie algebras
There is a classical correspondence between Lie algebras (over $\mathbb{R}$ or $\mathbb{C}$) and Lie groups in the finite dimensional case: to every Lie group $G$ there is an associated Lie algebra $\...
- 2,388
1
vote
0
answers
92
views
The $U({\frak g})v$-module generated by a single element of a $U({\frak g})v$-module
Let $\frak{g}$ be a finite dimensional complex semisimple Lie algebra and let $U(\frak{g})$ be its universal enveloping algebra. Take $V$ an infinite dimensional module over $U(\frak{g})$. Let $v \in ...
5
votes
0
answers
128
views
Classification of connected finite affine type A crystals
In the survey https://www.aimath.org/WWN/kostka/crysdumb.pdf the following statement is stated as a Conjecture 4.5 (due to Kashiwara): "Every connected affine crystal graph is isomorphic to a ...
- 163
3
votes
0
answers
95
views
Reference of general version of the PBW theorem and its consequences
Let $A$ be a commutative ring with identity and $L$ be a Lie algebra which is also a free module over $A$. I have seen the following statements:
The universal enveloping algebra $U(L)$ is isomorphic (...
- 1,713
3
votes
1
answer
369
views
The adjoint representation of $U_q({\frak sl}_2)$ on itself
Let $U_q(\frak{sl}_2)$ denote the quantum universal enveloping algebra of $\frak{sl}_2$, and consider the adjoint action
$$
\mathrm{ad}_X: U_q({\frak sl}_2) \to U_q({\frak sl}_2), ~~ Y \mapsto S(X_{(...
- 8,534
3
votes
1
answer
255
views
How does the constancy of an operator’s eigenvalues imply the integrability of its eigenvector distribution?
I am reading a paper, Yano and Ishihara, “Submanifolds with Parallel Mean Curvature Vector” (MSN), where the authors have constructed a linear operator, say $A$, on vector fields. They claim that ...
CommunityBot
- 1
4
votes
1
answer
129
views
Complexifications of minimal parabolic subalgebras
Let $\mathfrak{g}$ be a real semisimple Lie algebra with complexification $\mathfrak{g}_\mathbb{C}$. Recall that a parabolic subalgebra in $\mathfrak{g}_\mathbb{C}$ is one which contains a Borel ...
- 954
3
votes
1
answer
195
views
double shuffle lie algebra
I have a question about the definition of the double shuffle lie algebra discussed in section 1.3 of Sarah Carr's thesis (see https://www.imj-prg.fr/theses/pdf/sarah_carr.pdf)
Recall the definition ...
- 518
6
votes
2
answers
401
views
Relations between $3j$-symbols and intertwiners
I am trying to understand the relation between Wigner's $3j$-symbols (or Clebsch-Gordan coefficients) and matrix coefficients of intertwiners. I am new to this topic and need some help to understand ...
- 63.9k
4
votes
0
answers
214
views
Is the natural map from the free Lie algebra to the free associative algebra injective?
$\newcommand{\im}{\operatorname{im}}$Given a set $X$ and non-zero unital commutative ring $R$, let:
\begin{align}
A &= \mbox{free unital, associative algebra on $X$ with coefficients in $R$},\\
...
- 13k
3
votes
1
answer
245
views
Can non-geometrically reduced reduced subschemes happen for reductive groups?
The title is meant to be punchy, but also a tongue-in-cheek acknowledgement of the prevalence of ‘reduce’-derived words in this area. (Unfortunately, I overlooked the fact that the question in the ...
- 13k
1
vote
2
answers
311
views
Pullback of Lie algebras [closed]
Let $k$ be a field of characteristic 0 and let $\varphi:\mathfrak{g}\rightarrow\mathfrak{f}$ and $\psi:\mathfrak{h}\rightarrow\mathfrak{f}$ be maps of Lie algebras. Is there a reference showing that ...
- 2,821
4
votes
0
answers
262
views
de Rham Bloch-Ogus theory in positive characteristic
In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
- 2,054
5
votes
0
answers
171
views
Finite simple groups of automorphisms of finite simple Lie algebras
I begin by briefly recalling some basic facts in order to pose my question in context.
According to the classification, the finite simple groups are cyclic of prime order, are alternating on $n \geq 5$...
- 63.9k
13
votes
1
answer
411
views
Representations of $U_q(\mathfrak{sl}(2))$ as differential / difference operators
$\mathfrak{sl}(2)$ (over $\mathbb{C}$) with basis $E_\pm, H$ with commutation relations
$$
[H,E_{\pm}]=\pm 2 E_\pm,\quad [E_+,E_-]=H
$$
admits the well-known representation on $\mathbb{C}[x]$ with
$$
...
- 7,746
2
votes
1
answer
216
views
When does a finite irreducible Coxeter Group act on the cosets of a parabolic subgroup faithfully?
Let $(W,S)$ be a finite and irreducible Coxeter Group. For $J \subseteq S$, let $W_J = \langle s | s \in J \rangle$, a parabolic subgroup. For which $J$ is the action (group multiplication on the left)...
- 3,740
3
votes
0
answers
117
views
What total orders have people studied on Coxeter Groups?
I'm aware of the ShortLex total order that gives rise to the usual normal form. But are there any others that have naturally arose and people have studied?
- 193
9
votes
2
answers
481
views
Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra?
In a Lie algebra $\mathfrak{g}$ the Jacobi identity $\newcommand{\bracket}[2]{\left[#1\,#2\right]} \bracket{x}{\bracket{y}{z}} + \bracket{z}{\bracket{x}{y}} + \bracket{y}{\bracket{z}{x}} = 0$ holds. ...
2
votes
0
answers
408
views
What is a "Lefschetz SL2"?
In the paper "On Minuscule Representations and the Principal SL2" by B.H. Gross (link: here) and some others the terminology "Lefschetz $\operatorname{SL}_2$" is used. I think I am ...
- 2,821
11
votes
1
answer
617
views
Is there a unique "natural" action of $\mathsf{SL}_{n+1}$ on $\mathbb{R}^n$?
Context
By acting naturally via $\mathsf{SL}_3$ on $\mathbb{RP}^2=\{[x:y:z]\}$ and by taking the induced action on the affine hyperplane $z=1$ (which we identify with $\mathbb{R}^2$), one can realize ...
- 108k