All Questions
Tagged with gr.group-theory lie-algebras
126 questions
5
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Lattices in semisimple Lie groups
I am interested in the following question: Can semisimple Lie group of real rank $\geq 2$ contain an abelian lattice?
- 631
5
votes
2
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504
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A finiteness property for semi-simple algebraic groups
Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any semi-...
- 199
5
votes
1
answer
319
views
Malcev completion of free groups
Let $K$ be a field with $\operatorname{char} K=0$, $\hat{L}_n$ the complete free Lie algebra of $n$ variables $x_1,\dotsc,x_n$ and $\exp(\hat{L}_n)$ its associated group with the product given by BCH ...
- 985
5
votes
1
answer
2k
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Weyl groups of $E_6$ and $E_7$
The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...
- 951
5
votes
1
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316
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Identification problem: Does this group have a name?
I've encounter a group with properties that are very familiar, but I cannot say what group is it.
Consider the variables $(t,x,y,z)$, an affine transformation $M \in A(3)$ on the last three variables ...
- 690
5
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1
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820
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Maximal subgroups of semisimple Lie groups
The problem of finding and classifying the maximal subgroups of simple Lie groups like $SU(3)$ is well known and solved in the literature. What about maximal subgroups of semisimple groups like $SU(3) ...
user22139
5
votes
1
answer
163
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Goldman Lie algebra of a bordered surface vs. a closed surface?
How are the Goldman Lie algebra of a closed surface $\overline{S}$ and the bordered surface $S$ obtained by taking $\overline{S}$ and removing an open disc (or more generally, $n$ disjoint discs) ...
- 1,981
5
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1
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247
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Local vs global nilpotence class (Lazard correspondence)
The Lazard Correspondence is often phrased (for simplicity) for $p$-groups of nilpotence class $c < p$, but it works more generally whenever every 3-generated subgroup has nilpotence class $< p$,...
- 3,230
5
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1
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481
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Centralizer of hermitian matrices with zero trace
In Quantum Physics one often has to deal with commutators.
Here I want to denote by $H_0$ the set of all hermitian matrices with trace equal to zero!
One can easily relate it to $\mathfrak{su}(N)=...
5
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1
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552
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Invariant Laurent polynomials under cyclic group action
Start with the cyclic group $G:=\mathbb{Z}/p$ of prime order $p$ and and an integer lattice $P:=\mathbb{Z}^p$. Let $G$ act on $P$ by cyclic permutation of coordinates. There is an induced action on ...
- 333
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983
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Determining the Lie algebra elements exponentiating to the center of a Lie group
For a semi-simple compact Lie group $G$ with center $Z(G)$, one can characterize the preimage of $Z(G)$ in the Cartan subalgebra under the exponential map as the nodes of the Stiefel diagram (see for ...
- 1,582
5
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1
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273
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'Lie correspondence' for formal power series in non-commuting indeterminates
This is related to an earlier question of mine. I would like an argument or a reference (or a missing hypothesis if needed) for the following.
Let $\mathbb{F}\langle\langle \alpha\rangle\rangle$ and ...
- 1,089
5
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0
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171
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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 $n \geq 5$...
- 177
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123
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Conjugacy classes of plane k-jet group
Define $G(n, k)$ as a subgroup of $\rm{Aut}(\Bbb C[[x_1, \dots, x_n]]/\mathfrak m^{k+1})$ with identity linear part (so, group of $k$-jets of selfmaps of $\Bbb C^n$). I'm interested in the map from $G(...
- 4,600
5
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148
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Are there torsion-free restricted simple Lie algebras?
It is known that a torsion-free group can be simple (see e.g. Rataggi's paper https://www.degruyter.com/abstract/j/jgth.2007.10.issue-3/jgt.2007.028/jgt.2007.028.xml). I would like to know if the ...
- 99
4
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2
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1k
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Sum of all root lengths in simple Lie algebra
Part of one of my calculations involves (the innocent looking) expression
$\sum_{\alpha\in\Sigma} (\alpha,\alpha)$
for simple Lie algebras.
I have two methods of calculating it -- which don't agree. ...
- 461
4
votes
1
answer
243
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Second cohomology of the adjoint representation
Let $p$ be a prime and let $M_p$ be the $\mathrm{GL}_2(\mathbb{F}_p)$-module of $2 \times 2$ matrices over $\mathbb{F}_p$ with trace $0$ (the action is by conjugation).
Is it true that for $p$ large ...
- 11.3k
4
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1
answer
436
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Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie algebra: a reference request
Let $G$ be a (connected) semisimple algebraic group over an algebraically closed field $k$ of characteristic 0.
We consider the adjoint representation
$$ {\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$
...
- 14.2k
4
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1
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214
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The question about elementary equivalence of free products
Let $A,B,C,D$ be algebraic systems and $A$ and $B$ be elementary equivalent as well as $C$ and $D$. Are free products of $A,C$ and $B,D$ elementary equivalent if
$A,B,C,D$ are groups, or
$A,B,C,D$ ...
- 41
4
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2
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1k
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Dimension of Unipotent Radicals
A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subseteq G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic ...
4
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1
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171
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Which Lie groups admit finite generation by a set of Lie algebra elements? And what are some known choices of generators which realize this?
Consider a (finite-dimensional) real connected Lie group $G$ with Lie algebra $\frak{g}$. Take a generating set $\mathcal{G} = \{ X_1, \cdots X_n \} $ of $\frak{g}$, i.e. such that any element of $\...
- 143
4
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214
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Restricted Burnside Problem: Lower bound nilpotency class
Let $p$ be a prime and let $F$ be a free group of rank $d\geq 1$.
Kostrikin [1] proved that the $d$-generated Burnside group $B=B(d,p)=F/F^p$
of exponent $p$ has a maximal finite quotient
$\overline{B}...
- 1,991
4
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1
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1k
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Reductive Lie Groups and Complexification
Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, $G$...
- 4,920
4
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213
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The Weyl group of Kac-Moody algebra and Coxeter group
Let $\mathfrak{g}$ be a Kac-Moody algebra, $W$ be its Weyl group, generate by fundmental reflections {$r_1,...,r_n$}. For $i,j$, we have relation $(r_ir_j)^{m_{ij}}=1$, $1\le i,j\le n$. Let $W^{'}$ be ...
- 653
4
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144
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When does finite presentability of the associated graded Lie algebra of a group imply the group is finitely presented?
Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite ...
- 373
3
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3
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420
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Form of elements of a Lie algebra
How can we write the elements of a free Lie algebra? Is there any reference concerning this subject?
- 87
3
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2
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748
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For a Weyl group, what is the connection between its exponents and lengths of its elements?
The following seems to be true: if $|W_q| := \sum {q^{l(w)}}$, where the sum is taken over the elements $w$, then $|W_q| = \prod {(1 + q +...+ q^{e_i})}$, where the product is taken over the exponents ...
- 55
3
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2
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2k
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Casimir operator of a given Lie algebra and relation with its matrix representation
I'm following Gilmore's recipe to compute the abstract Casimir operator of a given algebra (in this example, I refer to algebra su(2)). This recipe bring up a matrix representation of the algebra and ...
- 255
3
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2
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418
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Homomorphism from noncompact semisimple Lie group to compact Lie group
Is it true that there is no homomorphism from a semisimple Lie group without compact factor to a compact Lie group?
3
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1
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140
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Is it true that all abelian p-nilpotent restricted Lie algebras are mirrors of finite abelian p-groups?
Let $k$ be a perfect field of characteristic $p$. A $p$-nilpotent restricted Lie algebra $\mathfrak{g}$ is a Lie algebra with $[p]$-restriction mapping such that $\mathfrak{g}^{[p]^n} = 0$ for some $n ...
- 542
3
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1
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361
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A functor that comes from a morphism in a bigger category
My loose question is like this: what would you say about an equivalence of categories where both are concrete categories, and the equivalence functor is induced from a set-theoretic bijection at the ...
- 7,320
3
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1
answer
137
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Subalgebras with finite codimension
In group theory it is well-known that every subgroup of finite index contains a normal subgroup of finite index. It is not true in general that for Lie algebras every subalgebra of finite codimenslon ...
- 379
3
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1
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298
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In what way do exact sequences of Lie ideals integrate to the category of groups?
Please excuse, very naive question:
Suppose $g$ is a topological Lie algebra over Q and $G$ = $exp(g)$ the associated group
(take free group on formal symbols $exp(X)$, X $\in$ $G$, and impose all ...
- 904
3
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0
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85
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Why do most eigenspaces of a Lie algebra automorphism have finitely many orbits?
I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper on the Local Langlands Conjectures (omitting the "well-known" proof).
Suppose $G$ is a ...
- 815
3
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0
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164
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Semi-direct products and associated graded Lie algebras
Let $G$ and $H$ be groups and $G\ltimes H$ their semi-direct product given by $f\colon G\to \operatorname{Aut}(H)$ satisfying $f(g)=\operatorname{id}_H$ in $H/[H,H]\,$ for all $g\in G$. In this ...
- 985
3
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0
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117
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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
3
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0
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98
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Semisimple subgroup of Euclidean group
Let $G$ be a closed and connected semisimple subgroup of the Euclidean group $E(n)$ (the group of isometries of $\mathbb R^n$).
Can we prove that $G$ is conjugate to a subgroup of $O(n)$?
- 2,535
3
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0
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122
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It there a nice way to describe the structure of Malcev-complete groups?
Let $\mathbb k$ be a field of characteristic zero. The grouplike functor $\mathbb G$ from complete Hopf algebras to groups is a faithful functor. Its image is the category of Malcev-complete groups ...
- 273
3
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255
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Roots of exceptional complex reflection groups
I am looking to do a case-by-case check of a conjecture I have about Shephard groups, which are a subclass of complex reflection groups. These were classified by Shephard and Todd and there is one ...
- 273
3
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0
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141
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Examples of divisible Lie algebras
We say that a nonzero Lie algebra $L$ is divisible, if for all elements $a$ and $b$ with $a\neq 0$, there exists $x\in L$ such that $[a, x]=b$. What are examples of divisible Lie algebras?
- 2,233
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359
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Does Branching in the Weight Diagram affect an embedding?
All groups here are compact semisimple Lie groups. Out of laziness I will use $B_7$ to mean $Spin(15)$.
Suppose that one has a group $H$ and a subgroup $G$. The embedding determines the decomposition ...
- 5,191
3
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0
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423
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Cohomologies associated to residually torsion-free nilpotent groups
This question is related to my previous question: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.
A group $G$ is ${\it residually \ torsion \ free \ ...
- 373
2
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1
answer
495
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Is every closed subgroup of $\text{GL}_n(K[[x]])$ finitely generated?
Let $n \in \mathbb{N}$, $K$ a finite field. Denote by $K[[x]]$ the (profinite) ring of formal power series over $K$. Note that $\text{GL}_n(K[[x]])$ is a profinite group.
Is every closed subgroup of $...
- 11.3k
2
votes
3
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318
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Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra
Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...
- 159
2
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1
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248
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What is the Molien series of the SO(2)-invariant ring on the plane (sometimes written C[X]^{SO(2)} )?
Let SO(2) be the group of rotations in the plane. What is the Molien series (sometimes called the Hilbert-Poincare series) of the SO(2)-invariant ring of polynomials?
N.B. The main goal being to ...
- 21
2
votes
1
answer
1k
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A question about flag variety of $SL(n,\mathbb{C})$
We know that the flag variety $SL(2,\mathbb{C})/B$ which $B$ is Borel subgroup, can be identified with $\mathbb{P^1}$, What can we say about $SL(n,\mathbb{C})/B$ which $B$ is Borel subgroup of $SL(...
user21574
2
votes
1
answer
316
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Decomposition of Lorentz-like groups
When studying the Lorentz group $O(1,3)$, one can decompose it into four parts... physicist usually called these
Proper-orthochronuos $\mathscr{L}^{\uparrow}_+$,
Proper-asynchronous $\mathscr{L}^{\...
- 690
2
votes
2
answers
367
views
Nilpotency of Lie Algebra from Structure Constants
Suppose we have a Lie algebra with structure constants
$$\mathrm{d}e^i=\sum_{j<k}a_{ijk}e^j\wedge e^k$$
for some coefficients $a_{ijk}$.
In this setting, how may be checked (perhaps ...
- 2,091
2
votes
1
answer
191
views
Normalizer of SU$(2)$ in SU$(6)$
Consider the $\mathfrak{su}(2)$ subalgebra of $\mathfrak{su}(6)$ embedded as
$$\mathfrak{su(2)}=\text{Span}\{\mathbb{1}_3 \times \sigma^i\}, \quad i=1,2,3$$
with $\sigma^i$ the Pauli matrices and $\...
- 155
2
votes
1
answer
250
views
Embedding of the adjoint group into $\mathrm{GL}(\mathfrak{g})$
Given a connected Lie group $G$ with corresponding Lie algebra $\mathfrak{g}$, the adjoint representation/action $\mathrm{Ad} : G \to \mathrm{GL}(\mathfrak{g})$ induces a Lie group homomorphism. It's ...
- 1,432