All Questions
6,486 questions
3
votes
1
answer
162
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Compact symmetric spaces and sub-root systems
Given two semisimple complex Lie algebras $\frak{g}$ and $\frak{n}$ such that the root system of $\frak{n}$ arises as a sub-root system of the root system of $\frak{g}$, does this then imply that $\...
4
votes
2
answers
227
views
Arithmetic application: Complete group ring and group ring for infinite group
Let $G$ be a profinite(infinite) group, $\Lambda$ be the complete group ring(Iwasawa algerbra) of $G$ over a unity ring $R$. My first question is that do we know something about the relation with $\...
- 319
1
vote
1
answer
112
views
Coprime polynomials and polynomial substitution
Let $F$ be a field, and let $P(X_1,\dots,X_m)$, $Q(X_1,\dots,X_m) \in F[X_1,\dots,X_m]$ be two coprime polynomials. Consider $n$ new polynomials $R_1(Y_{1,1},\dots,Y_{1,n}) \in F[Y_{1,1},\dots,Y_{1,n}]...
- 640
5
votes
1
answer
156
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The inverse limit of a sequence of ring surjections commutes with taking difference subsets of the respective units & gluing in some primes?
Define $R_n := \Bbb{Z}/p_n\#$ the ring of integers modulo primorial $p_n\# = p_n p_{n-1} \cdots p_1$. Let $U_n$ denote the group of units modulo $p_n\#$ in these rings.
Then if $f_{n,n+1}: \Bbb{Z}/p_{...
- 260
12
votes
3
answers
794
views
The orders of the exceptional Weyl groups
Who first calculated the orders of the Weyl groups $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$? How were these orders calculated?
- 417
1
vote
0
answers
35
views
Is it possible to manipulate heat kernel on H-type groups?
In Nathaniel Eldredge's work see here, he uses the explicit expression of the Heat Kernel on H-type groups (for example the Heisenberg group is an H-type group):
$$p_t(x,z)= (2\pi )^{-m} (4 \pi )^{-n}...
- 677
19
votes
2
answers
792
views
Topology on a module over a topological ring
The questions
Let $R$ be a topological ring, and let $M$ (with no topology) be an $R$-module. Does $M$ somehow "inherit" a topology from the action of $R$?
Here's a proposal for such a ...
- 41.4k
2
votes
1
answer
314
views
A reductive group is the complexification of a compact subgroup even if not connected?
The definition of a linear algebraic complex reductive group is sometimes using the connectedness hypothesis for the complex algebraic group sometimes not.
Here I use the following definition : a ...
- 1,128
10
votes
1
answer
243
views
If $E_\text{sep}/F$ is normal, then must $E/F$ be normal?
This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
- 452
0
votes
1
answer
187
views
Quotient of a ring by a left ideal
This is a simple algebra question I'm struggling with.
Let $A$ be a ring (with unity) and $I\subset A$ a left ideal and $B\subset A$ a two sided Ideal.
$A/I=B$ and $A/B=I$ (in the category of left $A$...
- 71
4
votes
0
answers
87
views
Doubling constructions beyond classical groups: general principles?
The doubling method for constructing integral representations of L-functions has been successfully applied to classical groups, as demonstrated in this paper. However, extending this method to a wider ...
- 111
0
votes
0
answers
124
views
Do the following two notions of quantum groups sometimes coincide?
On the one hand there is the notion of quantum groups due to Drinfeld and Jimbo. In there notion a quantum group is defined as a deformation of the universal enveloping algebra of a semisimple Lie ...
- 163
15
votes
3
answers
1k
views
Are automorphisms of matrix algebras necessarily determinant preservers?
Is every automorphism $\phi : A \to A$ of a subalgebra $A \subseteq M_n$ necessarily a determinant preserver?
I would assume that the answer is no in general, but I'm unable to find an example (or any ...
- 261
4
votes
1
answer
222
views
Recent research on polynomial identities
I work in computational complexity, where I work on the problem of polynomial identity testing over arithmetic circuits. One particular case is when the variables over the polynomial ring don't ...
- 49
2
votes
1
answer
211
views
Direct product of direct sum of a flat module
In the book "Rings and Categories of Modules" by Anderson & Fuller, this problem is given: If $V^A$, i.e. the direct product of the module $V$ by the index set $A$, is flat for all sets $...
- 355
3
votes
1
answer
100
views
Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces
Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, ...
4
votes
0
answers
100
views
Embedding of a nilpotent algebraic group in upper triangular matrices
Suppose we have a polynomial group law on $G=\mathbb{R}^n$ which gives it a structure of a nilpotent algebraic group.
Is it true that there exists an embedding of $G$ into the group of upper-...
- 728
0
votes
2
answers
208
views
Real matrix rings and associative hypercomplex numbers
Are there real matrix rings which are not hypercomplex number systems? Is there a canonical form of a real matrix ring?
By a hypercomplex number system I mean a finite-dimensional, unital, associative ...
- 21
4
votes
0
answers
158
views
Wedderburn-Malcev principal theorem for graded-finite algebras
Let $k$ be a field and $A$ be a noncommutative $k$-algebra with Jacobson radical $J$. If $A$ is finite-dimensional, the Wedderburn-Malcev says that $A$ has a subalgebra $S$ such that $$A = S \oplus J$$...
- 492
5
votes
2
answers
443
views
Series of discrete groups with a Lie group limit
The groups ${\mathbb Z}_N$ may be viewed as a series, $N=1,2,3,\ldots$, which in the limit $N\to\infty$ approaches $U(1)$. I realize this is a bit hand waving but I'm pretty sure it can be made ...
- 1,150
3
votes
0
answers
134
views
When do quotients of $G$-vector bundles exist?
Let's work in the category of smooth (paracompact, Hausdorff) manifolds. Let $M$ be a manifold and $G$ a Lie group acting on $M$. Suppose $E$ is a $G$-vector bundle on $M$ (that is, $G$ acts on $E$ by ...
- 51
2
votes
0
answers
46
views
Are two notions of power-associativity equivalent for loops?
According to Groupprops, a magma $X$ is called power-associative if for every element $x\in X$ there exists a sequence $(x^n)_{n\in\mathbb N}$ of elements of $X$ such that $x^1=x$ and $x^m\cdot x^n=x^{...
- 41.8k
2
votes
0
answers
80
views
Does restricting the eigenvalues of Hermitian matrices to the interval $[0,2\pi)$ make the exponential map to the unitary group bijective?
Let $U \in U(n)$ be a generic unitary matrix. Since the unitary group $U(n)$ is compact and connected, I know that the exponential map is surjective, i.e. that every $U \in U(n)$ has the form $U = e^{...
2
votes
2
answers
87
views
Computation of ideal of functions, given by explicit quadratic equations, vanishing on $G/P$ for the exceptional Lie group $G_2.$
In Section 10.6.6 of Procesi's "Lie Groups" he writes that a theorem due to Kostant tells us that for an algebraic group $G$ and a parabolic subgroup group $P,$ the ideal of functions ...
- 101
6
votes
1
answer
206
views
What makes the surreals special among other surreal-like fields?
Pre-setup: Let $\kappa = \aleph_\xi$ be an uncountable regular cardinal. Its role in this question is merely to sidestep the technical difficulties surrounding the (imho quite uninteresting) notion ...
- 32.5k
2
votes
0
answers
40
views
Reconstruction of a Poisson-Lie group structure from a Lie bialgebra $\mathfrak{g}$
Let $(\mathfrak{g}, [,], \delta)$ be a Lie bialgebra where $\delta$ is the cobracket. It is well-known that there exists a simply connected Poisson-Lie group $G$ such that $\mathfrak{g} = \mathrm{Lie}(...
- 255
1
vote
1
answer
87
views
An example of a commutative ring which is not SIP
Recall that a module $M_R$ ($R$ is a unital ring) is called an SIP-module if the intersection of any two summands of $M$ is a summand. The ring $R$ is called (left) right SIP-ring if the module (${}...
- 324
2
votes
1
answer
144
views
Paper request: Graev's classification of SU(2,2) irreducible unitary representations
I am interested in Graev's paper in "M. L. Graev:Dokl. Akad. Nauk SSSR,98, 517 (1954); Amer. Math. Soc. Transl.,66, 1 (1968)." in which the irreducible unitary representations of SU(2,2) are ...
- 679
1
vote
1
answer
102
views
Multiplicities and double and triple tensor products of simple $\frak{g}$-modules
Given a complex simple Lie algebra $\frak{g}$ and a simple module $V_{\lambda}$ for some dominant weight $\lambda$. Consider the tensor product decomposition
$$
V_{\lambda} \otimes V_{\lambda} \simeq ...
- 417
2
votes
0
answers
97
views
Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$
Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
- 2,907
3
votes
0
answers
50
views
Root systems of maximally noncomact Cartan subalgebras
Let $G$ be a real reductive Lie group, and $K$ a maximal compact subgroup in $G$. Write $\mathfrak{g}$ for the Lie algebra of $G$, and $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the Cartan ...
- 951
6
votes
2
answers
794
views
Tensor algebra and universal enveloping algebra
Let $\mathfrak g$ be a Lie algebra which is not reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
- 673
2
votes
0
answers
157
views
Centre of centralisers in connected reductive groups
Let $G$ be a connected reductive group over an algebraically closed field. Let $T$ be a maximal torus and $x\in T$. Let $G_x$ denote the centraliser of $x$ in $G$.
Question: What is an explicit ...
- 2,751
3
votes
0
answers
36
views
Computing the truncations (“ancestors”) of a surreal number from its Hahn series representation (“normal form”)
If $x$ is a surreal number and $\alpha$ an ordinal, let us denote $T_\alpha(x)$ and call $\alpha$-truncation of $x$ the surreal number whose sign sequence is obtained by truncating the sign sequence ...
- 32.5k
2
votes
0
answers
85
views
Coherent states on compact abelian state spaces and complexification
First, to establish notation, let $T^*(M)$ denote the cotangent bundle of a manifold $M$. Let $\widehat{(-)}:= \hom_{\sf LCAbGrp}(-,\mathbb{T}):{\sf LCAbGrp}^{\sf op}\simeq {\sf LCAbGrp}$ denote the ...
- 169
2
votes
0
answers
59
views
Measure rigidity for higher-rank subgroup actions on homogeneous space
Let $G$ be a semisimple Lie group, $\Gamma$ a lattice in $G$, and $H$ a higher-rank subgroup of $G$ (e.g., a non-split Cartan subgroup). Let $\mu$ be an $H$-invariant and ergodic probability measure ...
6
votes
1
answer
199
views
Combinatorial type construction of the free operad
$\DeclareMathOperator\RT{RT}$I am reading the book "Algebraic operads" by J. L. Loday and B. Vallete. The authors have given a combinatorial construction of the free operad over an $\mathbb{...
- 229
3
votes
0
answers
161
views
Generalized dimension property for rings
My question is very basic, I am looking for a characterization (and name) of rings $R$ satisfying the following property $\star$.
For any $V, W$ two finitely generated $R$-modules such that $V\oplus W\...
- 223
5
votes
1
answer
168
views
Countably compact Boolean algebras versus distributivity
Let us say that a complete Boolean algebra $B$ is:
countably distributive when for any sequence $(I_n)_{n\in\mathbb{N}}$ of sets and any elements $(u_{n,i})_{n\in\mathbb{N},i\in I_n}$ of $B$ we have
$...
- 32.5k
3
votes
1
answer
290
views
How do you define the composition of two $\mathbb{S}$-modules?
I am reading the book "Algebraic Operads" by J. L. Loday and B. Vallete. I am stuck with the definition of composition of $\mathbb{S}$-modules in Sec-5.1.6, pg. 99. Below I have written down ...
- 229
3
votes
0
answers
83
views
Non-commutative Gorenstein Koszul algebras
I was wondering if there exists a characterization of finite-dimensional Gorenstein Koszul algebras in terms of their Koszul dual?
- 359
7
votes
1
answer
259
views
A name for the Weyl group of $\frak{so_{2n}}$
For the $A$-series the Weyl group is the symmetric group $S_n$. For the $B$ and $C$ series the Weyl group is the hyperoctahedral group $\mathbb Z_2 \wr S_n$.
A) Does the $D$-series Weyl group $S_n \...
- 417
5
votes
1
answer
226
views
Nontrivial extension of the action of complex hyperbolic group $H$ on $\mathbb{C}$
Inspired by this question about conjugation of reql analytic maps to a holomorphic function and with a group action view point we ask the following question.
The complex Lie group $H=\...
- 356
11
votes
2
answers
558
views
Whether an isotone bijection from a power set lattice to another sends singletons to singletons
By the work of Paul Cohen (on the continuum hypothesis), one can neither prove nor disprove from the axioms of ZFC that a bijection $f$ from the power set $\mathcal{P}(S)$ of a set $S$ to the power ...
- 10.5k
0
votes
1
answer
113
views
Constructing an adjunction between algebras and differential graded algebras
Fix a ring R. I am looking for a construction of the adjunction between R-algebras and differential graded R-algebras. I am looking for a reference which constructs the left adjoint to the functor ...
user534056
15
votes
6
answers
671
views
Why, conceptually, does the torus normalizer in $G_2$ split?
Background: Let $G$ be a complex reductive group, $T$ a maximal torus, $N$ the normalizer of $T$ in $G$, and $W = N/T$ the Weyl group. All in all, we have a group extension
$$ 1 \to T \to N \to W \to ...
- 815
16
votes
0
answers
188
views
Representation theory of Pin groups
I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point.
Proctor describes this rule for $m = 2$...
- 1,574
1
vote
1
answer
119
views
Proving that compact Hausdorff groups are cofiltered limits of compact Lie groups
What is the easiest way to show that a compact hausdorff topological group is a closed subgroup of a product of finite dimensional Lie groups?
Here are the relevant definitions:
Definition: (compact ...
user531936
3
votes
1
answer
85
views
Isogeny of compact Lie group with central circle
Suppose $G$ is a connected, compact Lie group and $S^1 \subset G$ is a central subgroup. Can I write $G$ as a quotient of a product group $$G=(S^1 \times H)/Z$$
where the $S^1$ factor maps onto the ...
- 959
1
vote
1
answer
75
views
Lemma of Harada and Sai on sums of modules with a "chain" of monomorphisms between them
I am trying to get a contradiction from the following set of hypotheses:
Let $R$ be a ring. Let $M$ be a direct sum of non-zero $R$-modules $M_1$, $M_2$, $\dotsc$. For each $i\ge1$, let $f_i:M_i\to M_{...
- 1,644