All Questions
3,659 questions
0
votes
1
answer
60
views
Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 631
6
votes
0
answers
90
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
107
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 ...
- 6,622
-1
votes
0
answers
115
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
6
votes
1
answer
161
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 ...
- 63
5
votes
1
answer
231
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 $...
- 6,622
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 ...
- 21.8k
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 ...
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}...
- 1,101
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
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. ...
- 985
8
votes
1
answer
437
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 $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
11
votes
0
answers
427
views
Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
- 631
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 ...
- 6,622
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
2
votes
0
answers
92
views
Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
0
votes
0
answers
61
views
Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO.
I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
- 195
3
votes
1
answer
112
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
4
votes
1
answer
441
views
Large(ish) finite non-abelian subgroups of $\operatorname{GL}_n \mathbb C$ for $n>70$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SU{SU}\newcommand{\C}{\mathbb{C}}$My question is about large order finite non-abelian subgroups of $\GL_n\C$ without an ...
- 1,150
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 $\...
- 133
1
vote
0
answers
18
views
Behavior of the number of components of disconnected reductive groups when intersecting a Levi subgroup
Let $G$ be a connected reductive group over $\mathbb{C}$. Let $P=MN$ be a parabolic subgroup of $G$ with its Levi decomposition ($N$ the unipotent radical, $M$ a Levi).
Let $H\subset M$ be a finite ...
- 709
6
votes
0
answers
349
views
Quantum Hilbert's fifth problem
Hilbert's fifth problem inquires whether every locally Euclidean group is necessarily a Lie group. Von Neumann demonstrated that this is indeed true for the compact case.
The definition of a quantum ...
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
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
8
votes
1
answer
322
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
6
votes
3
answers
551
views
Conjecture about commutative semigroups
Conjecture: given any commutative semigroup $S$ of order $n \ge 4$, there exist $a, b \in S$ with $a \ne b$, an integer $m \ge \lfloor (n-1)/2 \rfloor$, and two $m$-element subsets $X = \{x_1, \ldots, ...
- 1,000
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
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
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
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
3
votes
0
answers
250
views
Action (of a graded monoid) required
Reference request: Did the construction below appear anywhere before? Any mentions of it or especially any links to something commonly known would be really helpful. I feel that it might be related to ...
- 321
6
votes
2
answers
502
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
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
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
123
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
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
98
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
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
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
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