All Questions
6,486 questions
2
votes
1
answer
198
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Maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$
After asking this question and finding this relevant paper, I would like to ask the following question:
For every $a,b \in \mathbb{C}$, denote:
$A_{a,b}=\mathbb{C}[(x-a)(x-b),x(x-a)(x-b),y]$
and
$B_{a,...
- 2,837
6
votes
3
answers
490
views
Boolean rings with many automorphisms
Does there exist an infinite Boolean ring $R$ (not assume unital, only associative) with the property that for any nonzero $x,y\in R$, there is a ring automorphism $\varphi\colon R\to R$ such that $\...
- 131
3
votes
1
answer
182
views
In dimension $n=5$, does a subgroup of $O(n)$ satisfying these properties exist?
I asked a question where @YCor provided a construction that seems to enable a group construction satisfying some properties when $n\ne 5$. However, in the case $n=5$, I am starting to think no such ...
- 185
11
votes
0
answers
283
views
Why are there so few irreducible admissible representations of $\text{GL}(n,\mathbb{R})$ (up to infinitesimal equivalence)?
Studying Langlands's classification of irreducible admissible representations, I have been rather stunned by the following:
Theorem
Up to infinitesimal equivalence, all irreducible admissible ...
- 511
2
votes
0
answers
73
views
On a possible generalization of heat kernel semigroups on Lie groups
Let $G$ be a compact matrix Lie group with Haar measure $\mu$. Then the heat kernel $\rho: G\times (0,\infty) \rightarrow \mathbb{R}$ satisfies
(1) $\rho(g_1g_2,t)=\rho(g_2g_1,t)$ for all positive $t$,...
- 505
4
votes
1
answer
239
views
True or false? Every left or right cancellative, duo semigroup is cancellative
A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
- 10.5k
5
votes
1
answer
883
views
Is this ring isomorphic to a quotient of a group algebra?
Consider the quotient of the free algebra $\mathbb{Q}\langle \alpha, \beta, \gamma, \delta, \varepsilon, \zeta \rangle$ by the two-sided ideal $I$ subject to the relations $$ \alpha\delta=\delta\alpha=...
- 1,093
11
votes
0
answers
183
views
Are algebras with rational structures dense in varieties of real Lie nilpotent algebras?
One says that a real nilpotent Lie algebra has $\mathbb Q$-structure if it has a basis with rational structure constants. It is well known that there are nilpotent Lie algebras without $\mathbb Q$-...
- 723
1
vote
0
answers
60
views
Reference for Gröbner-Shirshov algorithm in free restricted Lie algebras
I am searching for a reference on the Gröbner-Shirshov algorithm specifically for free restricted Lie algebras. I have already consulted the textbook by Bokut et al (Gröbner–Shirshov Bases Normal ...
- 1,013
8
votes
1
answer
321
views
Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
1
vote
0
answers
46
views
The difference between two description of affine Weyl groups
I have a question about the difference between two description of affine Weyl groups.
Let me write two descriptions of affine Weyl groups:
Let $\mathfrak{g}=\mathfrak{g}(A)$ be affine Lie algebras ...
- 137
1
vote
0
answers
89
views
The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
8
votes
2
answers
596
views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
5
votes
0
answers
216
views
Lifting a morphism between quasi-projective varieties
Let $\mathcal{V}$ be an affine algebraic variety over $\mathbb{R}$, $G$ be a finite group acting freely on $\mathcal{V}$. Consider the quotient space $Y:=\mathcal{V}/G$, which itself is a quasi-...
- 364
15
votes
1
answer
518
views
Pairs of matrices for which traces of powers are independent of the order
Let $A,B$ be $n\times n$ matrices over ${\mathbb C}$ such that, for all $m,k$ and all partitions $(i_1,\ldots ,i_r)$ of $m$ and $(j_1,\ldots ,j_r)$ of $k$ (perhaps with some zero parts),
$${\rm tr}\, (...
- 1,336
7
votes
2
answers
488
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
9
votes
2
answers
865
views
Multiplication in Peter-Weyl theorem
$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
- 2,964
4
votes
0
answers
267
views
If $\mathbb{C}[a,b,c] \subsetneq \mathbb{C}[x]$, then there exist $f,g$ s.t. $\mathbb{C}[a,b,c] \subseteq \mathbb{C}[f,g] \subsetneq \mathbb{C}[x]$
I ran into this MSE question and would like to ask about its answer and plausible generalizations.
The quoted MSE question asks if the following claim is true or false and why:
Claim: Let $a,b,c \in \...
- 2,837
0
votes
0
answers
98
views
An application of the Gleason-Montgomery-Zippin Theorem
In the book How groups grow by Avinoam Mann, the author cites the following theorem attributed to Gleason-Montgomery-Zippin.
Theorem 6.4 (Gleason–Montgomery–Zippin: solution of Hilbert’s Fifth ...
- 1
0
votes
1
answer
223
views
On the Irreducibility of Cyclotomic polynomials
Let $F$ be any field with $\operatorname{Char} F=q$. Let $p$ be a prime such that $p\neq q$. Suppose $F$ has no $p$-th root of unity except $1$. Is it true that the cyclotomic polynomial $X^{p-1}+\...
- 494
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
1
vote
0
answers
52
views
How large can the normalizer of $\mathrm{Ad}(G)$ in $\mathrm{GL}(\mathfrak{g})$ be?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ad{Ad}$Let $G$ be a real Lie group with Lie algebra $\mathfrak g$ (say reductive/semisimple if it makes the question easier).
I am interested in ...
- 1,942
2
votes
1
answer
327
views
Completion of a local ring is noetherian (under some hypothesis)
I was reading the proof of Lemma 10.12 in this paper. In the second sentence, the following fact is used implicitly:
Let $(R,\mathfrak{m})$ be a commutative local ring. Let $\widehat{R}$ be its $\...
- 293
8
votes
1
answer
361
views
Invertible matrix with group ring coefficient
Before asking the question I do need
some notations.
$G$ a (torsion-free) group, $\mathbb{Z}^{´}=\mathbb{Z}[\frac{1}{2}]$
$R:= \mathbb{Z}[G]$, $R^{´}=\mathbb{Z}^{´}[G]$ group rings.
$Mat_{n}(R)$ the ...
- 223
2
votes
0
answers
73
views
Are idempotents in a nonnegatively graded algebra conjugate to homogeneous idempotents?
Let $k$ be a field (assume algebraically closed and $\operatorname{char}(k) = 0$ if it helps) and $R = \bigoplus_{i \geq 0} R_i$ a (unital associative) non-negatively graded $k$-algebra. Furthermore, ...
5
votes
0
answers
226
views
Cohomology of representation varieties and the Hochschild cohomology
Let $k$ be a field, $A$ a $k$-algebra, and $V$ a $k$-vector space. Then we can consider the representation varieties of $A$ on $V$: $\mathrm{Hom}_{k\textrm{-alg}}(A, \mathfrak{gl}(V))$ and $\mathrm{...
- 985
3
votes
1
answer
108
views
Roots of polynomial $\sum_{\sigma \in W} x^{l(\sigma)}$
Let $W$ be Weyl group of a root system $\Phi$ (of finite dimensional simple Lie algebra). For $\sigma\in W$, $l(\sigma)$ be the its length. Consider the following polynomial
$$P_\Phi(x) = \sum_{\sigma ...
- 528
11
votes
1
answer
331
views
A question on groups having a subgroup which fixes a vector in every irreducible representations
Given a finite group $G$, I am interested in finding a non-trivial proper subgroup $H$ of $G$ such that $\mathrm{Ind}_H^G\mathbf{1}$ contains all the irreducible representations of $G$, that is, ...
- 251
6
votes
1
answer
320
views
Sum of derivative of polynomial over its simple roots
Let $P$ and $Q$ be polynomials over $\mathbb C$, and $n\in\mathbb N$ be a positive integer. I'm interested in the root sums of the form
$$ \sum_{P(x)=0}\frac{Q(x)}{P'(x)^n},$$
where the sum runs over ...
- 99
4
votes
1
answer
101
views
K-types of a representation of the minimal Gelfand-Kirillov dimension
Let $G$ be a noncompact real simple Lie group not of Hermitian type, and $\mathfrak{g}_0$ its Lie algebra. Fix a maximal compact subgroup $K$ in $G$ with its Lie algebra $\mathfrak{k}_0$. Write $\...
- 951
5
votes
0
answers
122
views
Algebraic groups and formal group laws in characteristic p
In characteristic zero, there is a well-known equivalence between Lie groups, formal group laws and Lie algebras.
Let $p$ be a prime. The equivalence between Lie groups and Lie algebras has an ...
- 413
3
votes
1
answer
160
views
Embedding flag manifolds of real semisimple lie group
I want to know given a connected (maybe we can assume it to be simply connected or linear) real semisimple lie group $G$ and one of its maximal parabolic group $P$, how can we embed the flag variety $...
- 175
4
votes
0
answers
183
views
Does a dual basis for $U_h(\mathfrak{sl}_2(\mathbb{C}))$ exist?
Let $\mathcal{F}_h(\operatorname{SL}_2(\mathbb{C}))$ be the $\mathbb{C}[[h]]$-algebra generated by $a, b, c, d$ subject to the following relations:
\begin{align*}
& ac = e^{-h}ca, \quad bd = e^{-h}...
- 255
3
votes
2
answers
255
views
Is being graded commutative a necessary condition on $A$ such that $H^*(A)$ is commutative?
If we consider any differential graded algebra $A^\bullet$, then its homology is a graded algebra, since the tensor product interacts well with homology.
A sufficient condidtion for the homology to be ...
- 4,408
2
votes
1
answer
121
views
Semi-direct decomposition of a solvable Lie group
(This is a cross-post from this MSE question)
I am searching for a reference or proof to the following fact (asserted at the top of page 2 here).
Let $G$ be a connected, solvable Lie group. Then $G = ...
- 185
4
votes
0
answers
97
views
Let $ G $ be a Lie group and $ H $ a connected subgroup of $ G $. If $ N_G(H)/H $ is finite does that imply $ H $ must be closed in $ G $?
Let $ G $ be a Lie group and $ H $ a connected subgroup of $ G $. If $ N_G(H)/H $ is finite does that imply $ H $ must be closed in $ G $?
The assumption that $ N_G(H)/H $ is finite cannot be weakened ...
- 4,081
0
votes
0
answers
72
views
Question about action of exponential of Lie algebras (Faraut and Koranyi's book)
I'm trying to understand a statement on page 49 in the book "Analysis on Symmetric Cones" by Faraut and Koranyi.
The situation is as follows. Let $\Omega$ be a symmetric cone in $V$ and ...
- 633
1
vote
0
answers
58
views
When is a bimodule that is projective as a right and as a left module also projective as a bimodule
Are there practical criteria for determining when a bimodule that is projective as a right and as a left module is projective as a bimodule? Some illustrative examples of what goes wrong and what goes ...
3
votes
1
answer
129
views
Questions about the quotient of extended Weyl group and the isomorphism of extended Weyl group
When I am reading a paper An Algebraic Characterization of the Affine Canonical Basis by Beck, Chari, and Pressley, and I have some questions about some notations.
In the paper, we assume that $\...
- 137
2
votes
1
answer
307
views
Serre functors and global dimensions
Let $k$ be a field.
Let $\mathcal{C}$ be an abelian category (over $k$).
We say that $\mathcal{C}$ has a finite global dimension if there exists integer $n > 0$ such that
$$
\operatorname{Ext}^i(M,...
- 889
3
votes
0
answers
89
views
Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$
Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
- 10.5k
3
votes
0
answers
155
views
Efficient computation of $x^n \bmod g(x)$ in $\mathbb{F}_2[x]$
Related to this question. I wish to compute $x^n \bmod g(x)$ in $\mathbb{F}_2[x]$ for some fixed and known upfront $g$. This problem pops up in computing the 'pure' CRC function of a bit sequence of ...
- 139
3
votes
0
answers
117
views
Which sigma-ideals in a sigma-algebra are contained in an ideal of null sets?
Let $X$ be a Polish space and $\mathcal{B}(X)$ be the $\sigma$-algebra of Borel subsets of $X$. Given a Borel probability measure $\mu$ on $X$, we write $\mathcal{N}(\mu) := \{ B \in \mathcal{B}(X) : \...
0
votes
0
answers
59
views
Bimodule endomorphisms of a bimodule over a noncommutative ring
Let $R$ be a noncommutative ring and $M$ an $R$-$R$-bimodule that is projective as left $R$-module. We know that the bimodule of left $R$-module endomorphisms ${}_REnd(M)$ is isomorphic to the tensor ...
1
vote
1
answer
78
views
Shape of convex invariant sets in symmetric spaces
Let $G$ be a semisimple Lie group of rank one and let $\Gamma$ be a convex-cocompact, Zariski dense subgroup.
Let $X=G/K$ denote the symmetric space and $\partial X$ its visibility boundary.
Let $\...
- 460
1
vote
0
answers
72
views
Normalizer of connected subgroup contained in the Weyl group?
Let $ G $ be a simple Lie group. Let $ H $ be a connected subgroup of $ G $ such that $ N(H)/H $ is finite. In such a case, is $ N(H)/H $ always a subgroup of the Weyl group of $ G $?
For $ G=\...
- 4,081
3
votes
1
answer
231
views
Does a representation of the universal cover of a Lie group induce a projective representation of the group itself?
Suppose that $G$ is a connected Lie group, $\tilde{G}$ its universal cover, $p:\tilde{G}\to G$ the covering map. Does a representation $\rho$ of $\tilde{G}$ on a finite-dimensional vector space $V$ ...
- 3,115
3
votes
0
answers
75
views
Non-vanishing of a "push-forward" Fourier–Harish-Chandra transform on a compact set
Let $G \subset \operatorname{GL}_d(\mathbb{R})$ be a non-compact semi-simple Lie group and $K \subset G$ a maximal compact subgroup. Let $\mathfrak{g}$ (resp. $\mathfrak{k}$) be the Lie algebra of $G$ ...
- 81
8
votes
1
answer
685
views
The state of the art on topological rings - the Jacobson topology
I was recently studying the Jacobson density theorem and I found it quite interesting.
Most textbooks I've seen, including Jacobson's own Basic Algebra, only spend a few lines about the reason why it ...
- 183
16
votes
0
answers
218
views
If a map between unital rings preserves multiplication and successor, does it preserve addition?
Welcome to my first MathOverflow posting!
This is a question about rings, all of them assumed to be both unital and associative.
Let $f\colon R\to S$ be a map between rings such that $f(xy)=f(x)f(y)$ ...
- 161