All Questions
Tagged with gr.group-theory lie-algebras
126 questions
3
votes
2
answers
748
views
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
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
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
2
votes
1
answer
316
views
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
3
votes
3
answers
420
views
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
1
vote
0
answers
252
views
Generalizing groups via the Hall-Witt identity
In studying the integrability problem for Lie algebra representations, I have been led to wonder whether generalizing the notion of group by dropping associativity, while keeping the Hall-Witt ...
- 393
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
2
votes
1
answer
572
views
Free affine actions of Borel subgroups
Call an upper triangular $m\times m$ matrix $A$ admissible if the lowest non-zero entry of $A-I$ lies in the last column, and is strictly lower than any other non-zero entry of $A-I$. I'll also regard ...
- 1,089
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
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
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
1
vote
2
answers
465
views
subgroups with the same number of roots that the group.
When thinking in terms of Dynkin diagrams, I am naively used to see that the diagram for a subgroup can be extracted from the diagram of the group by removing some roots. Now, I noticed that for SO(10)...
- 437
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
2
votes
0
answers
866
views
dual Coxeter number, affine algebra, invariants under twisting
Sometime ago we came across invariant quantities under twisting of all affine algebra. (See the appendix E of http://arxiv.org/abs/hep-th/0403076 .) Choose the convention so that the longest root has ...
- 191
1
vote
1
answer
1k
views
Ado's Theorem Proof
Could someone please tell me what I am missing in the following argument. Either my understanding of the exact statement of Ado's theorem is wrong, or there is a flaw in my argument below.
For a ...
- 1,161
-2
votes
2
answers
734
views
Lie algabra of symmetric group
It's easy to see that the descending central series of a group induces a graded Lie algebra .(see for example Serre's Harvard lectures or Magnus-Solitar book). I think in general this can be ...
- 1,180
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
3
votes
0
answers
359
views
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
votes
0
answers
423
views
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
3
votes
1
answer
361
views
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
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
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
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
4
votes
2
answers
1k
views
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
3
votes
1
answer
298
views
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
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