All Questions
6,486 questions
2
votes
0
answers
62
views
An elementary proof of the equivalence of the Moufang identities
By a well-known result of Bol (1937) and Bruck (1958), for any loop the following two identities are equivalent:
B: $x(y(xz))=((xy)x)z$
M: $(xy)(zx)=(x(yz))x$.
A proof of the equivalence (B)$\...
- 41.8k
6
votes
0
answers
86
views
Generic representations of $\mathrm{GL}_n(\mathbb{R})$
Let $F$ be a local field of characteristic $0$, $G=\mathrm{GL}_n(F)$.
When $F$ is $p$-adic, Bernstein and Zelevinsky classified the irreducible generic representations. The statement is:
Let $\delta_{...
- 709
7
votes
0
answers
102
views
Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights
$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
- 63.9k
-1
votes
0
answers
114
views
Stability of flow map
$\DeclareMathOperator\Diff{Diff}$Setting:
Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...
- 5,405
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 ...
- 7,853
-3
votes
0
answers
140
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
4
votes
1
answer
251
views
Connected Frobenius algebras non-semisimple as an object
A Frobenius algebra object $A$ in a tensor category $\mathcal C$ is said to be connected if $\text{Hom}_{\mathcal C}(\mathbb{1}, A)$ is a one dimensional vector space, where $\mathbb {1} $ denotes the ...
10
votes
1
answer
604
views
Is it possible to average a riemannian metric over an action and preserve curvature bounds?
Let $M$ be a finite dimensional smooth manifold endowed with a riemannian metric $g$ and a smooth action $\mu$ by a compact Lie group $G$. Averaging $g$ over $G$ defines a new metric
$$g'(X,Y)=\int_Gg(...
- 45k
5
votes
1
answer
230
views
Explicit Jacquet-Langlands correspondence for real reductive groups
Let $G$ be a connected reductive group over $\mathbb R$. Let $G'$ over $\mathbb R$ be an inner form of $G$ with ${}^LG={}^LG'$. By local Langlands correspondence over $\mathbb R$, if a $L$-packet of $...
- 761
4
votes
1
answer
324
views
The Hausdorff codimension of singular matrices vs. the Hausdorff codimension of points with divergent trajectories
Let $G:=SL(m+n,\mathbb R)$ and $\Gamma :=SL(m+n,\mathbb Z)$ and $X:=G/\Gamma$.
(1) Let $M$ denote the set of all $m \times n$ matrices with real entries. A matrix $A \in M$ is called $\textit{singular}...
CommunityBot
- 1
6
votes
1
answer
158
views
Centralizers in semisimple Lie group
For a semisimple complex Lie algebra $\mathfrak{g}$ and a regular element $X\in \mathfrak g$ the centralizer of $X$ in $\mathfrak g$ is a Cartan subalgebra (see Knapp, 'Lie Groups beyond an ...
- 12.9k
3
votes
1
answer
304
views
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$ as $\operatorname{GL}(n,\mathbb{C})$-modules
Consider the natural representations of $\operatorname{GL}(n,\mathbb{C})$ in the spaces
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$.
Is it known how to ...
- 12.9k
2
votes
0
answers
46
views
Are maps between cohomology of homogeneous vector bundles morphisms of representations?
Let $X = G/P$ a rational homogeneous variety, e.g. a grassmannian. Consider a short exact sequence $$ 0 \longrightarrow E_1 \longrightarrow E_2 \longrightarrow E_3 \longrightarrow 0$$
where $E_i$ are ...
- 41
3
votes
1
answer
102
views
Which compact Lie groups have an upper bound on the dimension of irreducible continuous representations?
To fix notation, if $G$ is a compact Lie group, $Rep(G)$ denotes the set of continuous irreducible unitary representations of G, and $\widehat{G}$ denotes the quotient $Rep(G)/\sim$, which identifies ...
- 8,597
1
vote
1
answer
187
views
Relating classic spectral decomposition with Euclidean Jordan algebras
I'm currently getting into studying optimization problems over symmetric cones (NSCP) and I'm having some trouble to understand something.
Let me first give some context, sorry if it is repetitive to ...
CommunityBot
- 1
5
votes
1
answer
653
views
What are the maximal closed subgroups of $ \operatorname{SU}_3 $?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$What are the maximal closed subgroups of $ \SU_3 $?
This question is inspired by Lie subgroups of SU(3). Interesting partial answers to that ...
- 4,081
0
votes
0
answers
45
views
Artinian simple algebras with involution
Let $A$ be an Artinian simple $K$-algebra with involution $*$. The algebra $A$ has a primitive idempotent $e$, and $D_e:=eAe$ is a division $K$-algebra due to Schur's lemma and the isomorphism $(eAe)^{...
- 143
4
votes
1
answer
363
views
Is there a "natural" interpretation of the power function for octonions and for sedenions?
This question is a sequel to Is there a definition of $\log(x)$ for quaternion/octonion $x$? Since $\log(x)$ is multivalued even for complex $x \in \mathbb{C}$, it is impossible to define $\log(x)$ ...
CommunityBot
- 1
7
votes
1
answer
177
views
Ergodicity of action of finite index subgroups in the boundary
Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}...
- 12.3k
2
votes
0
answers
126
views
A property of matrices formed by pairing of roots and coroots
Let $A$ be an $n\times n$ integral matrix, define its level $l(A)$ as
$$l(A) := \begin{cases}0 &\det A = 0 \\ \text{smallest integer } N \text{ such that } NA^{-1} \text{ is integral} &\det A \...
- 528
8
votes
1
answer
436
views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 716
3
votes
1
answer
110
views
Lie subalgebra annihilated by all derivations
Let $k$ be a field and $\mathfrak{g}$ a Lie algebra over $k$. Put $K(\mathfrak{g}) = \bigcap_{f\in\mathrm{Der}(\mathfrak{g})} \mathrm{Ker}(f)$, which is a Lie subalgebra of $\mathfrak{g}$.
Question. ...
- 12.9k
2
votes
1
answer
153
views
On generation of $A_n$ by elements of prime order
There is a question regarding generation of finite simple groups with elements of prime order. Recently, Guralnick, Shareshian, Woodroofe and Teräväinen made advances in this direction. We have, for ...
- 22.5k
3
votes
0
answers
170
views
Cellular structure of $F_4$
Is there the cellular structure of the Exceptional Lie group $F_4$?
Is there a reference to it?
Thanks
1
vote
1
answer
80
views
Hilbert symbol of a quaternion algebra given ramified places
I am reading the paper: https://projecteuclid.org/journals/experimental-mathematics/volume-17/issue-3/Derived-Arithmetic-Fuchsian-Groups-of-Genus-Two/em/1227121388.full
in order to find an explicit ...
- 3,009
2
votes
1
answer
149
views
Baer sums of extensions
Apologies in advance if this question is too basic, I looked briefly through Weibel and the stacks project and couldn't find any relevant reference.
Let $\mathcal{A}$ denote an abelian category, and ...
- 2,907
3
votes
1
answer
114
views
Selmer complex and total complex
Thanks for your reading. I'm studying Selmer complex book by Jan Nekovar. For the definition of Selmer complex I meet a problem.
In the introduction(page 9, 0.8.0) the author gives us a definition of ...
- 10.9k
13
votes
2
answers
1k
views
What's the deal with De Morgan algebras and Kleene algebras?
The notion of Boolean algebras, and the corresponding classical propositional logic, is very standard, and it is easy to find information about them (for example, among many other such works, there is ...
- 47.1k
6
votes
0
answers
102
views
Computer program for free restricted Lie polynomial
I am conducting numerical experiments involving the Gröbner–Shirshov Basis for restricted Lie algebras. At each step of the computation, I need to work with restricted Lie polynomials. Specifically, I ...
- 63.9k
5
votes
1
answer
179
views
Semisimplicity of algebras in fusion categories
Let $\mathcal{C}$ be a fusion category and $A \in \mathcal{C}$ be an algebra object. We say that $A$ is semisimple if its category of (right) modules $\mathsf{mod}_A(\mathcal{C})$ is a semisimple ...
- 61
4
votes
0
answers
236
views
Jacobian of exponential map
I am playing around with the coarea formula and came across the problem of finding the Jacobian of the exponential map.
Let $G$ be a compact, semisimple Lie group with associated Lie algebra $\...
2
votes
3
answers
181
views
Stabilizers of the action of Levi on abelianization of nilpotent radical
$\DeclareMathOperator\Lie{Lie}$Let $G$ be a simple connected reductive group over $\mathbb C$. Consider a parabolic subgroup $P=MU$ of $G$, where $M$ is a Levi of $P$ and $U$ is the unipotent radical ...
- 2,516
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 ...
- 38.6k
3
votes
1
answer
103
views
References: rigorous algorithms for elementary computations in base-b with complexity estimates
Definitions/Notation: Fix positive integers $b$ and $M$. Consider the set of real numbers which can be exactly expressed with $2M+1$ coefficients in base $b$, defined by
$$\mathcal{X}(b,M):=\{x\in \...
- 63.9k
1
vote
0
answers
71
views
Component groups of stabilizers for linear representations
Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$.
Given a vector $v \in V$, it is natural to consider its stabilizer group $...
- 6,622
7
votes
2
answers
2k
views
Geodesics equation on Lie groups with left invariant metrics
First of all, I am so sorry if this question is not appropriate to be here. I tried to ask something similar on Math Stack Exchange but it didn't have much attention. Any comment and I delete the ...
- 126
2
votes
0
answers
93
views
Free, easy-to-use program for noncommutative algebra over finite fields
I am looking for a computer program that can handle computations in noncommutative algebra over a finite field of prime order $p$.
My requirements are:
The program should be free, as I do not have ...
- 63.9k
20
votes
4
answers
3k
views
Online References for Cartan Geometry
I would like to learn more about Cartan Geometry ("les espaces généralisés de Cartan"). I ordered Rick Sharpe's book "Differential Geometry: Cartan's generalization...", which would take a long time ...
- 26.3k
36
votes
1
answer
3k
views
Whence “homomorphism” and “homomorphic”?
Today homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen?
“Homomorphic” (and “homomorphism” as “property of being ...
- 31.5k
2
votes
1
answer
239
views
n-ary (polyadic) group "defined for tuples"
Consider an n-ary (polyadic) group, which is a generalization of the "usual" ("binary") groups to n-tuples. The definition requires appropriate standard generalizations of ...
CommunityBot
- 1
4
votes
1
answer
364
views
Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$
Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread.
Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...
CommunityBot
- 1
4
votes
2
answers
366
views
Notion of prime congruences
We have the idea of a prime ideal in a commutative ring $R$ but in universal algebra, we generalize the notion of ideal to that of a congruence. I’ve thought over the question of what a prime ...
CommunityBot
- 1
0
votes
1
answer
294
views
Hopf algebras actions
Can you write down a general type of Hopf algebra action? How do you justify the name "action", when it is already used for group actions?
There must be a common core, if the same term is ...
CommunityBot
- 1
1
vote
0
answers
153
views
Does it make sense to take a formal power series over a non-ring? [closed]
Suppose we have the set of nonnegative integers. This is obviously not a ring, and not even a group. But are we able to identify ring-like properties for it if we take a formal power series over it?
...
- 12.9k
0
votes
0
answers
32
views
Right maximal ideals in skew-Laurent rings over division Rings
Let $R$ be a Noetherian domain, and let $D$ denote its division ring. Define $S = D_q[x_1^{\pm 1}, x_2^{\pm 1}]$ as an iterated skew-Laurent polynomial ring with the relation $x_1 x_2 = q x_2 x_1$. Is ...
- 63.9k
3
votes
1
answer
111
views
Generalization of a result of Kostant related to Gauss decomposition and Toda lattices
I found myself needing a generalization of a result of Kostant in his famous paper
B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math, Volume 34, 1979, ...
- 173
0
votes
0
answers
92
views
Finitely generated module over a skew Laurent polynomial ring $\mathbb{K}[x_1^{\pm 1},x_2^{\pm1}]$
Let $A := \mathbb{K}_{\Lambda}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}, x_4^{\pm 1}]$, $B := \mathbb{K}_{\Lambda'}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}]$, and $C := \mathbb{K}_{\Lambda''}[x_1^{\pm 1}, ...
- 923
-1
votes
1
answer
803
views
Question on an exercise on homological algebra?
Suppose $R$ has finite global dimension $n$, $N$ is a finitely generated module, $F$ is a free module, and $\operatorname{Ext}^n(N, F) \neq 0$. Then $\operatorname{Ext}^n(N, R)$ is also non-trivial.
...
- 113
5
votes
2
answers
1k
views
An example where finitistic dimension does not equal right global dimension?
The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup ...
- 113
0
votes
0
answers
61
views
$\mathcal{R}$ is finite over $L_0[e,A]$
Let $\sigma:L \longrightarrow L$ an automorphism of infinite order, where $L_0 \subset L$ is its fixed field. Let $R$ be a commutative subring of $L\{\sigma\}$. Let
\begin{equation}
\mathcal{R}=\...
- 69