All Questions
6,486 questions
149
votes
7
answers
23k
views
Homotopy groups of Lie groups
Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
- 4,014
145
votes
14
answers
50k
views
Why study Lie algebras?
I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
- 2,751
123
votes
18
answers
14k
views
How do you decide whether a question in abstract algebra is worth studying?
Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...
114
votes
2
answers
12k
views
How would you solve this tantalizing Halmos problem?
$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
Geometric ...
- 4,736
106
votes
3
answers
10k
views
Has the Lie group E8 really been detected experimentally?
A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced,
"Quantum ...
- 20.7k
96
votes
16
answers
18k
views
Why is it a good idea to study a ring by studying its modules?
This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules ...
- 118k
95
votes
11
answers
6k
views
Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?
Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...
- 2,600
94
votes
2
answers
7k
views
$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$
Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
- 63.1k
87
votes
5
answers
10k
views
When is $A$ isomorphic to $A^3$?
This is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a ...
- 63.1k
81
votes
26
answers
7k
views
What would you want on a Lie theory cheat poster?
For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, ...
78
votes
7
answers
8k
views
Example of a manifold which is not a homogeneous space of any Lie group
Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth ...
- 8,559
77
votes
7
answers
21k
views
What is the symbol of a differential operator?
I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....
- 54.6k
76
votes
9
answers
6k
views
Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
The question is the extent to which we can unify addition
and multiplication, realizing them as terms in a single
underlying binary operation. I have a number of questions.
Is there a binary ...
- 236k
74
votes
1
answer
6k
views
$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$
Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
- 63.1k
73
votes
6
answers
7k
views
Surprisingly short or elegant proofs using Lie theory
Today, I was listening to someone give an exhausting proof of the fundamental theorem of algebra when I recalled that there was a short proof using Lie theory:
A finite extension $K$ of $\mathbb{C}$...
71
votes
28
answers
8k
views
Results from abstract algebra which look wrong (but are true)
There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...
65
votes
6
answers
9k
views
Origin of terms "flag", "flag manifold", "flag variety"?
These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably ...
- 52.9k
64
votes
4
answers
8k
views
What is the current status of the Kaplansky zero-divisor conjecture for group rings?
Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good ...
- 1,065
62
votes
25
answers
70k
views
Linear Algebra Texts?
Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...
62
votes
5
answers
10k
views
Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I can ...
- 11.2k
61
votes
3
answers
7k
views
Why is there no Cayley's Theorem for rings?
Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a ...
- 3,230
60
votes
8
answers
13k
views
Why the Killing form?
I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...
- 7,273
59
votes
4
answers
15k
views
Group theory in machine learning
I'm a Machine Learning researcher who would like to research applications of group theory in ML.
There is a term "Partially Observed Groups" in machine learning theory which has been ...
- 607
56
votes
2
answers
5k
views
A condition that implies commutativity
Let $R$ be a ring. A notable theorem of N. Jacobson states that if the identity $x^{n}=x$ holds for every $x \in R$ and a fixed $n \geq 2$ then $R$ is a commutative ring.
The proof of the result for ...
- 8,842
55
votes
0
answers
2k
views
What did Gelfand mean by suggesting to study "Heredity Principle" structures instead of categories?
Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the ...
- 1,238
53
votes
5
answers
8k
views
Beautiful descriptions of exceptional groups
I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
- 1,422
53
votes
5
answers
5k
views
Does this formula have a rigorous meaning, or is it merely formal?
I hope this problem is not considered too "elementary" for MO. It concerns a formula that I have always found fascinating. For, at first glance, it appears completely "obvious", while on closer ...
- 15.3k
53
votes
7
answers
14k
views
Good lattice theory books?
A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - ...
52
votes
7
answers
8k
views
"Algebraic" topologies like the Zariski topology?
The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...
52
votes
2
answers
5k
views
Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?
$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The ...
- 541
51
votes
1
answer
2k
views
Invertible matrices over noncommutative rings
Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples?
The question popped up ...
- 2,941
51
votes
2
answers
3k
views
$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group
Recently, prompted by considerations in conformal field theory, I was lead to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free.
...
- 43.2k
48
votes
7
answers
15k
views
Classification of (compact) Lie groups
I would like to study/understand the (complete) classification of compact lie groups. I know there are a lot of books on this subject, but I'd like to hear what's the best route I can follow (in your ...
47
votes
9
answers
11k
views
What are the reasons for considering rings without identity?
I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.
47
votes
10
answers
6k
views
Algebraic theorems with no known algebraic proofs
What are some good examples of algebraic theorems that have no known algebraic proofs?
A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...
47
votes
2
answers
3k
views
Non isomorphic finite rings with isomorphic additive and multiplicative structure
About a year ago, a colleague asked me the following question:
Suppose $(R,+,\cdot)$ and $(S,\oplus,\odot)$ are two rings such that $(R,+)$ is isomorphic, as an abelian group, to $(S,\oplus)$, and $...
- 7,207
45
votes
8
answers
6k
views
What makes a theorem *a* "nullstellensatz."
I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic ...
44
votes
2
answers
2k
views
Fermat's Last Theorem for integer matrices
Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
- 1,889
43
votes
18
answers
5k
views
Results in linear algebra that depend on the choice of field
Linear algebra as we learn it as undergraduates usually holds for any field (even though we usually learn it for the complex, or real, numbers).
I am looking for a list of concepts, and results, in ...
43
votes
3
answers
7k
views
transcendental Galois theory
Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ of $K \setminus L$, ...
- 65.4k
42
votes
9
answers
6k
views
Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?
Harold Williams, Pablo Solis, and I were chatting and the following question came up.
In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...
42
votes
6
answers
7k
views
An algebra of "integrals"
When discussing divergent integrals with people, I got curious about the following:
Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace)
$$\int_0^{\infty}: ...
- 3,806
42
votes
4
answers
33k
views
What is the intuition for the trace norm (nuclear norm)?
I will word this question in terms of linear operators acting on $\mathbb{C}^n$ for simplicity. Feel free to provide an answer in terms of more general Hilbert spaces if you think it makes more sense ...
- 539
41
votes
7
answers
6k
views
Why don't ideals and quotients work well for categories?
Ideals are intimately related to quotients and congruence relations. They clearly play a very important role in ring theory and order theory. So do normal subgroups in group theory. (Enriched) ...
- 3,937
41
votes
3
answers
3k
views
What is the classifying space of "G-bundles with connections"
Let $G$ be a (maybe Lie) group, and $M$ a space (perhaps a manifold). Then a principal $G$-bundle over $M$ is a bundle $P \to M$ on which $G$ acts (by fiber-preserving maps), so that each fiber is a $...
- 54.6k
40
votes
9
answers
10k
views
Simplest examples of rings that are not isomorphic to their opposites
What are the simplest examples of
rings that are not isomorphic to their
opposite rings? Is there a science to constructing them?
The only simple example known to me:
In Jacobson's Basic Algebra (...
- 5,654
39
votes
5
answers
5k
views
When does a ring surjection imply a surjection of the group of units?
The following might be a very trivial question. If so, I don't mind it being closed, but would appreciate a reference where I could read about it.
Let $R$ and $S$ be commutative rings and let $R^\...
- 33.3k
39
votes
4
answers
5k
views
Is there a universal property for Witt vectors?
Do the Witt vectors satisfy a universal property?
- 8,004
39
votes
3
answers
4k
views
Can every Lie group be realized as the full isometry group of a Riemannian manifold?
Suppose a finite-dimensional Lie group $G$ is given. Does there exist a connected manifold $M$ and a Riemannian metric $g$, such that $G$ is the full isometry group of $(M,g)$?
For example if I try to ...
- 2,830
39
votes
5
answers
4k
views
Is there an explicit construction of a free coalgebra?
I am interested in the differences between algebras and coalgebras. Naively, it does not seem as though there is much difference: after all, all you have done is to reverse the arrows in the ...
- 9,132