All Questions
Tagged with gr.group-theory lie-algebras
126 questions
3
votes
0
answers
122
views
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
1
vote
0
answers
107
views
Weyl Group action on the complement of the Tits Cone in a Kac-Moody algebra
Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-...
- 31
3
votes
2
answers
2k
views
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
6
votes
1
answer
464
views
Adjoint orbits of a finite group of type $G_2$
Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{...
- 993
1
vote
0
answers
65
views
Is there a decomposition exists for $e^{c(K_++K_-)^2}$
In the usual $SU(1,1)$ group:
$$[K_+,K_-]=-2K_z,\quad [K_z,K_\pm]=\pm K_\pm.$$
Is there a decomposition exist for $e^{c(K_++K_-)^2}$?
Of course there won't exist a decomposition to $e^{K_+},e^{K_-},...
user120129
3
votes
1
answer
137
views
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
7
votes
1
answer
358
views
Lie algebra of a p-group
Given a p-group P, the first hochschild cohomology of the group algebra (over a field of characteristic p) of P is a nonzero Lie algebra. Is it known what Lie algebra results depending on P? I have no ...
- 26.5k
5
votes
2
answers
1k
views
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
4
votes
1
answer
243
views
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
5
votes
1
answer
163
views
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
12
votes
1
answer
2k
views
Relationship between the Witt algebra and vector fields on the circle
I have seen some (apparently contradictory) claims by mathematicians and physicists regarding the existence of an (infinite dimensional) Lie group whose Lie algebra is the Virasoro algebra.
The ...
- 1,329
13
votes
2
answers
515
views
Free groups and free restricted Lie algebras
If $G$ is any group and $\gamma_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows
$$\mathcal{L}(G) = \bigoplus_{k=1}^{\infty} \gamma_k(G) / \...
- 44.8k
3
votes
0
answers
255
views
Roots of exceptional complex reflection groups
I am looking to do a case-by-case check of a conjecture I have about Shephard groups, which are a subclass of complex reflection groups. These were classified by Shephard and Todd and there is one ...
- 273
1
vote
0
answers
201
views
Laplacian on two Lie groups have the same Lie algebra
I know that if $G$ is a Lie group and $\mathfrak g = span\{X_i, 1\leq i \leq n\}$ be its Lie algebra, where $\{X_i\}$ are the vector fields of $G$. Then, the Casimir-Laplacian of $G$ is given by
$$\...
- 650
30
votes
0
answers
999
views
Follow-up to Steinberg's problem (12) in his 1966 ICM talk?
Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (...
- 52.9k
2
votes
3
answers
318
views
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
5
votes
2
answers
504
views
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
6
votes
1
answer
475
views
What is this Lie algebra?
Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero.
If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' ...
- 2,099
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
8
votes
1
answer
562
views
The parity of the full automorphism group order of finite non-abelian groups of prime exponent
Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?
- 3,738
15
votes
2
answers
838
views
factorization of the regular representation of the symmetric group
Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation.
Question: Does there exist a representation $V$ (of dimension $(...
- 2,537
5
votes
1
answer
481
views
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)=...
13
votes
1
answer
2k
views
Some questions about the Malcev completion
Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n \mathbb{Q}...
- 4,484
1
vote
0
answers
189
views
Poincaré inequality for connected Lie groups
Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is:
symmetric,
adapted (in the sense that there is no proper subgroup $H$ such ...
- 527
4
votes
1
answer
214
views
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
0
votes
2
answers
1k
views
Representation Theory of $U(N)$
(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...
- 401
6
votes
1
answer
719
views
Torsion in profinite groups
Is there a finitely generated profinite group $G$ with a closed subgroup of infinite index $K \leq G$ such that for every $g \in G$ there exists some $n \in \mathbb{N}$ for which $g^n \in K$ ?
Can $G$...
- 11.3k
5
votes
1
answer
552
views
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
8
votes
3
answers
502
views
Polarizations generate the ring of invariants?
The symmetric group $S_n$ acts on $\mathbb R^n$ by permuting the coordinates and the ring of polynomial invariants is generated by the elementary symmetric polynomials. If we restrict the action to ...
- 187
5
votes
1
answer
2k
views
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
13
votes
1
answer
455
views
Variety of nilpotent Lie algebras or $p$-groups
Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either:
1) Let $\mathcal{L}$ ...
- 63.9k
11
votes
2
answers
1k
views
Sums of degrees of irreducible complex characters
The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant to determine the dimension of a maximal torus of the Lie algebra associated to ...
- 581
7
votes
3
answers
2k
views
Characterising the adjoint representation of SU(N)
One can show that the adjoint representation of $\mathrm{SU}(n)$, the image of the map $\mathrm{Ad}:\mathrm{SU}(n) \rightarrow \mathrm{Aut}(\mathrm{su(n)})\subset \mathrm{GL}(\mathrm{su}(n))$, is an $...
- 71
2
votes
1
answer
495
views
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
4
votes
0
answers
144
views
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
votes
0
answers
141
views
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
11
votes
2
answers
1k
views
Realizing a subgroup of a Lie group as a stabilizer subgroup
Let $G$ a compact semisimple Lie group, $H$ a subgroup of $G$. Is it always possible to find an irreducible representation $R$ of $G$ such that the stabilizer of an $x\in R$ is "locally isomorphic" to ...
- 679
5
votes
1
answer
983
views
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
10
votes
2
answers
2k
views
Chevalley Groups over an arbitrary ring.
My question is simply about the Chevalley groups over rings. In many books, including Carter's book on "Simple groups of Lie types", the groups are considered over fields. I have checked the ...
- 2,508
1
vote
1
answer
608
views
Para-Complexification of Lie Groups
Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
...
user21574
2
votes
1
answer
1k
views
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
0
votes
1
answer
274
views
when $g^*$ is invariant under $Ad(G)$?
Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra.
Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under $...
user21574
6
votes
2
answers
768
views
When did the meaning of the term "metabelian" change?
I just realised that the meaning of the term "metabelian", when applied to groups, or Lie algebras, seems to have changed over years. (These days, it means that $[[G,G],[G,G]]$ is trivial, while in ...
- 16.9k
9
votes
1
answer
1k
views
Easy argument for "connected simple real rank zero Lie groups are compact"?
Let $G$ be a connected simple Lie group. It is known that if $G$ has real rank zero, then $G$ is compact.
Background: every connected (semi)simple Lie group $G$ (with Lie algebra $\mathfrak{g}$) has ...
- 223
5
votes
1
answer
820
views
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
4
votes
1
answer
1k
views
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
13
votes
1
answer
760
views
Characteristic subgroup of nilpotent group that is not invariant under powering
I want an example of a nilpotent group $G$, a characteristic subgroup $H$, and a prime number $p$ such that:
$G$ is $p$-powered, i.e., every element of $G$ has a unique $p^{th}$ root in $G$.
$H$ is ...
- 7,320
4
votes
2
answers
1k
views
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 ...
1
vote
2
answers
341
views
Copies of ax+b inside the AN part of an Iwasawa decomposition?
As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
- 25.8k
6
votes
2
answers
2k
views
Dense subgroups of Lie Groups
SETUP: Let $G$ be a connected Lie group, and $H\subset G$ be a FINITELY GENERATED dense subgroup.
I am interested in knowing what kind of information one can infer on the complexity of $H$.
I am ...
- 1,452