All Questions
Tagged with gr.group-theory lie-algebras
126 questions
44
votes
2
answers
3k
views
What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?
Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$, and let $\mathfrak g \text{-rep}$ be its category of finite-dimensional modules. Then $\mathfrak g\text{-rep}$ comes equipped ...
- 54.6k
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
20
votes
1
answer
3k
views
Motivation for Hall-Witt identity
I've wondered for a while about the (Hall-)Witt identity in group theory:
$[[a,b^{-1}],c]^b \cdot [[b,c^{-1}],a]^c \cdot [[c,a^{-1}]],b]^a = 1$.
(Here, $x^y$ means $y^{-1}xy$ and $[x,y]$ means $...
- 489
18
votes
3
answers
1k
views
Is a retract of a free object free?
I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?
- 1,875
18
votes
1
answer
727
views
(Dis)similarity between groups and Lie algebras
There are many questions which sound similar or the same for groups and Lie algebras. Some (very few, actually) of those questions have identical solutions and answers. Some have identical answers but ...
- 2,804
15
votes
6
answers
671
views
Why, conceptually, does the torus normalizer in $G_2$ split?
Background: Let $G$ be a complex reductive group, $T$ a maximal torus, $N$ the normalizer of $T$ in $G$, and $W = N/T$ the Weyl group. All in all, we have a group extension
$$ 1 \to T \to N \to W \to ...
- 815
15
votes
2
answers
838
views
factorization of the regular representation of the symmetric group
Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation.
Question: Does there exist a representation $V$ (of dimension $(...
- 2,537
13
votes
4
answers
3k
views
What is a "block" in an abelian category?
In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which ...
- 52.9k
13
votes
1
answer
2k
views
Some questions about the Malcev completion
Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n \mathbb{Q}...
- 4,484
13
votes
2
answers
515
views
Free groups and free restricted Lie algebras
If $G$ is any group and $\gamma_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows
$$\mathcal{L}(G) = \bigoplus_{k=1}^{\infty} \gamma_k(G) / \...
- 44.8k
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
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
12
votes
2
answers
494
views
A specific coset decomposition of $\mathrm{GL}_n(\mathbb{C})$
Disclaimer: I am a theoretical chemist (not a mathematician). I have tried asking this question at Math SE with no luck (https://math.stackexchange.com/questions/4080696/a-specific-coset-decomposition-...
- 121
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
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
11
votes
2
answers
934
views
Non-isomorphic complex Lie groups with the same exceptional Lie algebra for $\mathfrak{g_2,f_4,e_6,e_7,e_8}$?
An exceptional complex Lie algebra is a simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five such Lie algebras: $\mathfrak{g}_{2}$, ${\mathfrak {f}}_{4}...
- 2,147
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
11
votes
3
answers
554
views
Uniform setting for computing orders of algebraic groups over finite quotients of the integers?
A couple of recent questions on MO have involved the characters or the orders of specific finite groups of the form $G(\mathbb{Z}/n\mathbb{Z})$ for a familiar algebraic group $G$ defined over $\mathbb{...
- 52.9k
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
10
votes
4
answers
998
views
Longest Element of an Affine Weyl Group
I know that the Weyl groups of affine Lie algebras don't have a longest element, but are there any good substitutes for w_0. In particular, is there any good substitute for a reduced decomposition of ...
- 5,548
10
votes
3
answers
1k
views
subgroup of SU(N) with maximal manifold dimension
Given the group SU(N) of NxN unitary matrices, does there exist a subgroup S
with a manifold dimension larger than the SU(N-1) manifold dimension and
smaller than the SU(N) one? S should not ...
- 1,207
9
votes
1
answer
373
views
Is the Magnus Lie algebra of a finitely presented group finitely presented
Let $G$ be a finitely presented group and let $L(G)$ be the Magnus Lie algebra associated to the lower central series of $G$. This $L(G)$ is a graded Lie ring generated by its degree 1 piece $L_1(G) =...
- 93
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
9
votes
1
answer
386
views
Different definitions of formality for groups
Let $X$ be a space with fundamental group $G$. Recall that the de Rham fundamental group of $X$ is the inverse limit of the Malcev completions of the nilpotent truncations of $G$. This has a Lie ...
- 383
9
votes
1
answer
230
views
Yang-Mills algebra and lower central series of surface groups
Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area.
First, in "...
8
votes
3
answers
1k
views
A reference for the Chevalley Groups
Hello everyone
I would like to learn basic theory of the Chevalley Groups. There are several references for this subject, like "Introduction to Lie algebras and representation theory" by Humphreys, ...
- 2,508
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
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
8
votes
0
answers
200
views
Logarithm of a $p$-group in $\mathrm{GL}_n(p)$
$\def\GL{\operatorname{GL}}\def\ZZ{\mathbb{Z}}\def\FF{\mathbb{F}}\def\Id{\mathrm{Id}}\def\fu{\mathfrak{u}}$Let $p$ be prime, let $n<p$, let $U_n(\FF_p)$ be the group of $n \times n$ upper ...
- 156k
8
votes
0
answers
1k
views
Computational complexity of multiplication in a nilpotent group?
What is the computational complexity of multiplication in a Carnot group ?
Background: A Carnot group is a real nilpotent Lie group $N$ whose Lie algebra $Lie(N)$ admits a direct sum decomposition
...
- 785
7
votes
1
answer
2k
views
If two Lie algebras are isomorphic, under which conditions will their Lie groups also be isomorphic?
Let $G$ and $G'$ be compact connected Lie groups (which are not necessarily simply connected) with Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$. Suppose that the two Lie algebras are isomorphic, ...
- 97
7
votes
3
answers
617
views
Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3...
Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem.
I wonder what is known/expected for char p=2,3 ?
More vague ...
- 24.9k
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
7
votes
2
answers
669
views
Élie Cartan's paper "Les groupes réels simples, finis et continus" of 1914
Question 1.
Does Élie Cartan's paper
Les groupes réels simples, finis et continus,
Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355
contain a classification of $\Bbb C$-linear involutions of simple ...
- 14.2k
7
votes
1
answer
259
views
A name for the Weyl group of $\frak{so_{2n}}$
For the $A$-series the Weyl group is the symmetric group $S_n$. For the $B$ and $C$ series the Weyl group is the hyperoctahedral group $\mathbb Z_2 \wr S_n$.
A) Does the $D$-series Weyl group $S_n \...
- 415
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
7
votes
1
answer
571
views
Does Aut(G) → Out(G) always split for a compact, connected Lie group G?
The outer automorphism group of a topological group $G$ is constructed by the short exact sequence
$$
1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \...
- 473
7
votes
1
answer
210
views
Universal enveloping algebra of Malcev Lie algebra associated to nilpotent group
Let $G$ be a finitely generated torsion-free nilpotent group. The Malcev completion of $G$ is a nilpotent Lie group $N$ into which $G$ embeds as a lattice. One way to construct this is to take the ...
- 153
7
votes
0
answers
179
views
Regarding $F_4$ and $G_2$ Lie algebras, do there exist $F_n$ or $G_n$ families of Lie algebras?
For example, $E_6$ exceptional Lie algebra is part of the $E_n$ series of Lie algebras (Kac-Moody algebras). Are $F_4$ or $G_2$ maybe also parts of some $F_n$ or $G_n$ series of Lie algebras or are ...
6
votes
2
answers
501
views
Group of diffeomorphisms and its tangent space i.e. its Lie algebra
So I feel like there are many questions and also many sources on what I am asking, but I still don't understand what I think is a very basic thing in my head:
It is known, that for a Lie group $G$ (...
- 63
6
votes
2
answers
400
views
Relations between $3j$-symbols and intertwiners
I am trying to understand the relation between Wigner's $3j$-symbols (or Clebsch-Gordan coefficients) and matrix coefficients of intertwiners. I am new to this topic and need some help to understand ...
- 1,429
6
votes
2
answers
2k
views
Dense subgroups of Lie Groups
SETUP: Let $G$ be a connected Lie group, and $H\subset G$ be a FINITELY GENERATED dense subgroup.
I am interested in knowing what kind of information one can infer on the complexity of $H$.
I am ...
- 1,452
6
votes
1
answer
446
views
Analog of the Lie Product formula for commutators
Let $X, Y$ be elements of a Lie algebra. Consider the group $G$ generated by (limits of) arbitrary products of the elements
$$ G = \langle{e^{tX},e^{sY}\rangle}$$
for all $t,s$. The Lie product ...
- 361
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
6
votes
3
answers
993
views
Occurrences of a simple reflection in the longest element of a Weyl group?
While looking at a preprint I've just bumped into a question about the longest element $w_0$ of a Weyl group $W$ (say irreducible of a Lie type $A$ - $G$ and of rank $n>1$, to simplify). ...
- 52.9k
6
votes
2
answers
476
views
Centralizer of a cyclic subgroup within the group algebra $\mathbb{C} S_N$ of the symmetric group
Let us take the group algebra $\mathbb{C} S_N$ and the subgroup $H=Z_N$ generated by the element
$\sigma=(123\dots N)$, which is a cyclic shift. What is the structure of the centralizer of $H$ within $...
- 293
6
votes
1
answer
308
views
"Almost-ideals" in the (simple) Lie algebra of an algebraic group?
Let $G<\mathrm{GL_n}$ be a simple linear algebraic group defined over a finite field $K$. Let $\mathfrak{g}$ be its Lie algebra. Assume $\mathfrak{g}$ is simple.
Is it necessarily the case that ...
- 20.2k
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
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
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