All Questions
723 questions
10
votes
4
answers
2k
views
Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$
One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$
the ring of ...
- 3,541
7
votes
0
answers
601
views
A few questions about $E_6$ and its symmetric spaces
Preface
The purpose of my question - on high level - is to understand exceptional symmetric spaces. My latest idea is to embed them into Lie group. There is quite nice embedding of 32-dimensional $E_{...
user21230
7
votes
0
answers
268
views
Generalising supercommutativity as a grading by the $1$-truncated sphere spectrum
A discussion that has been going recently is that supersymmetry corresponds to grading over the sphere spectrum, coming from an insight due to Kapranov.
To formalise such a statement, one needs a ...
- 11.8k
6
votes
1
answer
1k
views
Solid rings and Tor
A solid ring is a ring $R$ such that the multiplication
$R\otimes_{\mathbb{Z}} R \to R$ is an isomorphism.
These were classified by Bousfield and Kan; they are
subrings of $\mathbb{Q}$,
$\mathbb{Z}/...
- 12.5k
6
votes
1
answer
231
views
Does Manin's construction of non-commutative endomorphism algebra $\mathrm{End}(A)$ produce Koszul algebra, if $A$ is Koszul?
$\newcommand{\dual}{\mathrm{dual}}\DeclareMathOperator\End{End}\DeclareMathOperator\Fun{Fun}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\GL{GL}$Around 1986–7 Yu.Manin proposed natural and ...
- 24.9k
6
votes
0
answers
272
views
Exceptional symmetric spaces embedded in exceptional Lie group
In Yokota (1959) and Atsuyama (1977) papers one can find embedding of projective space $\mathbb OP^2$ into Lie group $F_4$. Lately I come to following idea to have embedding of all four projective ...
user21230
4
votes
0
answers
762
views
Rewrite sum of radicals equation as polynomial equation
My question is about a method described in [Dr.Math forum][1] for simplifying equations involving sums of radical functions.
(The following is a transcription of the example given by Dr. Vogler):
--- ...
- 153
4
votes
1
answer
614
views
About the conjugation of semi-simple subgroups
Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...
- 199
3
votes
1
answer
637
views
Faithful finite-dimensional unitary representations
Is there any characterization of the non-compact connected Lie groups that possess faithful finite-dimensional unitary representations?
0
votes
2
answers
357
views
Rank of a $ \mathbb{Z}_{p}[[T]] $ module
Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can ...
- 1,209
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
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
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 ...
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
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 - ...
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.
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 ...
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
38
votes
18
answers
24k
views
Learning about Lie groups
Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.
36
votes
3
answers
2k
views
Are large powers of polynomials linearly independent?
Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
- 4,653
34
votes
8
answers
4k
views
Uncountable counterexamples in algebra
In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
34
votes
1
answer
5k
views
Freyd-Mitchell's embedding theorem
Freyd–Mitchell's embedding theorem states that: if $A$ is a small abelian category, then there exists a ring R and a full, faithful and exact functor $F\colon A \to R\text{-}\mathrm{Mod}$.
I have been ...
- 3,004
33
votes
3
answers
6k
views
When is a finite dimensional real or complex Lie Group not a matrix group
I have a smattering of knowledge and disconnected facts about this question, so I would like to clarify the following discussion, and I also seek references and citations supporting this knowledge. ...
- 1,161
28
votes
4
answers
5k
views
Triality of Spin(8)
Among simple Lie groups, $Spin(8)$ is the most symmetrical one in the sense that $Out(Spin(8))$ is the largest possible group. A description of this outer automorphism groups is as follows. $Spin(8)$ ...
- 281
25
votes
8
answers
6k
views
What is the "right" definition of a ring?
This is somewhat related to Greg's question about groups and abelian groups. Suppose you met someone who was well-acquainted with groups, but who was unwilling to accept rings as a meaningful object ...
- 118k
24
votes
3
answers
2k
views
Real Lie groups versus real linear algebraic groups: differences in connexity and fundamental group
There are many introductory texts on real Lie groups, and many on linear algebraic groups in general, but fewer on the specific case of linear algebraic groups over the reals, and even fewer that try ...
- 32.5k
23
votes
3
answers
2k
views
How bad can $\pi_1$ of a linear group orbit be?
Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the ...
- 31.5k
23
votes
6
answers
5k
views
cohomology of BG, G compact Lie group
It has been stated in several papers that $H^{odd}(BG,\mathbb{R})=0$ for compact Lie group
$G$. However, I've still not found a proof of this. I believe that the proof is as follows:
--> $G$ compact ...
- 1,709
22
votes
6
answers
3k
views
Automorphism group of real orthogonal Lie groups
I would like to understand what is the "outer-automorphism group" $Out$ of $SO(p,q)$ and $O(p,q)$, where $p+q >0$ and $pq \neq 0$. My working definition of $Out$ is as follows:
Let us denote by $...
- 2,816
22
votes
1
answer
1k
views
Word maps on compact Lie groups
Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and ...
- 25.5k
21
votes
4
answers
5k
views
The number of ideals in a ring
Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here.
Let $R$ be a finite commutative ring with identity. Under what conditions the number of ...
- 1,881
20
votes
2
answers
1k
views
The first unstable homotopy group of $Sp(n)$
Thanks to the fibrations
\begin{align*}
SO(n) \to SO(n+1) &\to S^n\\
SU(n) \to SU(n+1) &\to S^{2n+1}\\
Sp(n) \to Sp(n+1) &\to S^{4n+3}
\end{align*}
we know that
\begin{align*}
\pi_i(SO(...
- 19.3k
20
votes
3
answers
2k
views
Can a module be an extension in two really different ways?
(Edit: I've realized that there was an error in my reasoning when I was convincing myself that these two formulations are equivalent. Hailong has given a beautiful affirmative answer to my first ...
20
votes
4
answers
3k
views
Relationship between the cohomology of a group and the cohomology of its associated Lie algebra
Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is $\...
- 373
20
votes
3
answers
9k
views
Curvature of a Lie group
Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
- 251
20
votes
3
answers
840
views
Is there an analogue of the hive model for Littlewood-Richardson coefficients of types $B$, $C$ and $D$?
If $V_\lambda$, $V_\mu$ and $V_\nu$ are irreducible representations of $\operatorname{GL}_n$, the Littlewood-Richardson coefficient $c_{\lambda\mu}^\nu$ denotes the multiplicity of $V_\nu$ in the ...
- 313
20
votes
2
answers
1k
views
Why do flag manifolds, in the P(V_rho) embedding, look like products of P^1s?
Bert Kostant mentioned an odd fact to me some time ago. As usual (with such statements), fix a
complex, connected, reductive) Lie group $G$, with maximal torus $T$, and Weyl vector $\rho$ equal to ...
- 27.8k
19
votes
2
answers
3k
views
Does every irreducible representation of a compact group occur in tensor products of a faithful representation and its dual?
(Previously posted on math.SE with no answers.)
Let $G$ be a compact Lie group and $V$ a faithful (complex, continuous, finite-dimensional) representation of it. Is it true that every (complex, ...
- 118k
19
votes
4
answers
2k
views
What is the geometric object corresponding to a subalgebra in a polynomial ring
Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to ...
- 1,961
19
votes
5
answers
2k
views
Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?
Let $G$ be a simple algebraic group group over $\mathbb C$.
Let $V$ be a self-dual representation of $G$.
Let $\lambda$ be the highest weight of $V$.
Write $\lambda$ as a sum of fundamental weights: $...
- 43.2k
18
votes
3
answers
702
views
Existence of a ring with specified residue fields
Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$?
To prevent things from being too easy, I require ...
- 706
18
votes
3
answers
3k
views
Which groups have only real and quaternionic irreducible representations?
Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:
1) it's not isomorphic to its dual (...
- 22.3k
18
votes
4
answers
2k
views
For which rings $R$ is $\mathrm{SL}_n(R)$ generated by transvections?
Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.
Under which conditions is the group $\operatorname{SL}_n(R)$ generated by transvections?
(A transvection is a matrix with $1$ ...
- 6,614
17
votes
2
answers
1k
views
In a compact lie group, can two closed connected subgroups generate a non-closed subgroup?
Let $H$,$K$ be closed connected subgroups of a compact Lie group $G$. Let $L:=\langle H,K \rangle$ be the subgroup they generate, ie, the smallest subgroup of $G$ containing them both. Must $L$ be ...
- 853
17
votes
6
answers
3k
views
What's an example of a transcendental power series?
Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$?
I am looking for elementary example (so there should be a proof of transcendence that does ...
- 3,284
17
votes
1
answer
3k
views
Differences in philosophy between Lie Groups and Differential Galois Theory
As far as I have heard,Sophus Lie's aim was to construct an analogue of galois theory for differential galois theory. I am familiar with lie group but not with differential galois theory. What is the ...
- 2,106
17
votes
3
answers
905
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
- 2,071
16
votes
0
answers
860
views
Is "being a full ring of quotients" a Morita invariant property?
Definition and context:
An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
- 2,454
16
votes
2
answers
1k
views
Which commutative groups are the group of units of some field?
Inspired by a recent question on the multiplicative group of fields. Necessary conditions include that there are at most $n$ solutions to $x^n = 1$ in such a group and that any finite subgroup is ...
- 118k