All Questions
Tagged with ra.rings-and-algebras reference-request
329 questions
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
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 $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
13
votes
2
answers
1k
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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 ...
- 32.5k
6
votes
0
answers
102
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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 ...
- 1,013
4
votes
0
answers
132
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Ring theoretical aspects of the DAHA
The double affine Hecke algebras (DAHA) were introduced by Cherednik in his study of Macdonald's inner product conjectures (which were solved affirmatively).
Nowdays there are many variations of the ...
- 3,318
2
votes
0
answers
76
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Does a matrix ring over a ring satisfy the Koethe conjecture if the coefficient ring itself satisfies the Koethe conjecture?
I just want to know whether the following statement is true or false.
If $R$ is a ring satisfying the Koethe conjecture, then the matrix ring over $R$ also satisfies the Koethe conjecture.
Or is it ...
5
votes
0
answers
112
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Finitely generated projective modules over Noetherian endomorphism ring
Let $\mathcal A$ be a locally Noetherian Grothendieck abelian category and $M\in \mathcal A$ be a Noetherian object. Set $B:=\text{End}_{\mathcal A}(M)$. Let $B$-mod be the category of finitely ...
- 413
4
votes
1
answer
239
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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
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
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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
89
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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
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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
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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
3
votes
0
answers
89
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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
8
votes
1
answer
685
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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
5
votes
1
answer
367
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Reference request: locally erasable delta-functor is universal
It is well-documented that an erasable delta-functor $(F^n,\delta)$ is universal. However, in 'small' abelian categories (in the technical sense or otherwise), there aren't always enough objects to ...
- 28.7k
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
2
votes
0
answers
119
views
gcrd and associates of an element of the quaternion algebra over a totally real number field $K$
Let $K$ be a totally real number field of class number 1, and $Q$ the quaternion algebra over the ring of integers of $K$ with basis
$\{1,i,j,k\}$ such that $i^2 = j^2 = k^2 = -1$ and $ij = -ji, ik = ...
- 133
1
vote
0
answers
28
views
Most general filtered algebras with Hilbert polynomials and multiplicities
Let $k$ be any base field and $A$ an affine infinite dimensional $k$-algebra.
Let $\mathcal{F}= \{ A_i \}_{i \geq 0}$ be a finite dimensional filtration for $A$: that is, $k \subset A_0$ and each $A_i$...
- 3,318
3
votes
0
answers
152
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My category is rigid: what this implies for representation theory?
I am studying a subcategory $\mathcal{C}$ of modules for an associative noncommutative algebra $A$ (which is in fact also a Hopf algebra).
It is clear from our definition of $\mathcal{C}$ that it is ...
- 3,318
5
votes
0
answers
141
views
Gelfand-Kirillov dimension and tensor products
$\DeclareMathOperator\GK{GK}$Let $k$ be the base field.
The Gelfand-Kirillov dimension was introduced by Gelfand and Kirillov in their seminal paper on the Gelfand-Kirillov conjecture.
A very famous ...
- 3,318
0
votes
0
answers
64
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Proof of a folkloric result about PI-algebras [duplicate]
I am not not an specialist in PI-algebras, but I can say I have a rather good understanding on the subject.
It is, of course, interesting to discover if an algebra $A$ is a PI-algebra. But it is also ...
- 3,318
4
votes
0
answers
119
views
Adjoining new factors for primes in UFDs
It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...
- 18.7k
6
votes
1
answer
239
views
Attempts to define a matrix exponential over (as much as possible) general fields
Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as
$$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$
where ...
- 361
4
votes
1
answer
133
views
Second cohomology group of the contact Lie algebra $K_3$
Let $F$ be a field of characteristic zero and, for all $n>0$, consider the contact Lie algebra $K_{2n+1}$. It follows from Corollary 3 of the paper [V. Guillemin - S. Shnider: Some stable results ...
- 630
3
votes
1
answer
385
views
Concrete examples of derived categories
What examples of abelian categories $\mathcal{A}$ are there such that the derived category $\mathcal{D}(\mathcal{A})$ can be described concretely? For example, is there a concrete way of describing $\...
- 1,474
2
votes
2
answers
77
views
Reference request for a subfamily of regular graphs
[Repost of same question math stack exchange which got no answers]
I'm looking for literature on the following family of graphs:
Call a regular graph $G=(V,E)$ (of regularity degree $d$) nice if there ...
- 21
0
votes
0
answers
56
views
Names for product-like algebras involving a "duo of directed pseudoforests"
I am looking for the names (and/or for any information regarding) two algebras, one "free" and one "restricted" by an equivalence class.
In both cases, there is an (infix) binary ...
2
votes
0
answers
91
views
A recursive description of the smallest divisor-closed subsemigroup containing a set
Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid).
...
- 10.5k
3
votes
1
answer
170
views
Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational
Let a (rational) Puiseux monoid be a non-trivial submonoid of the non-negative rational numbers under (the usual operation of) addition. It is not difficult to show that, if $f \colon H \to K$ is a (...
- 10.5k
3
votes
1
answer
233
views
Tangent space of a GIT quotient of $GL_{N}$
Let $G:=\operatorname{GL}_{N}$ act on its Lie algebra $\mathfrak{g}:=\mathfrak{gl}_{N}$ by conjugation. Then it acts naturally on the associated ring $\mathcal{O}(\mathfrak{g})$ of (algebraic or ...
- 131
1
vote
0
answers
64
views
Groups with prescribed Ulm invariants
In Kaplansky's book infinite abelian groups he provides (through some exercises) a complete classification of $p^{\infty}$-torsion countable abelian groups in terms of Ulm invariants. In other words ...
- 11
6
votes
0
answers
179
views
Elementary equivalence for rings
Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-...
- 3,318
4
votes
1
answer
341
views
Some folklore about crystaline rings of differential operators
This question is a follow up to my previous question on rings of crystaline differential operators, to which I refer for the adequate definitions.
First, let's consider the case of an algebraically ...
- 3,318
2
votes
2
answers
432
views
Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)
I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here.
I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...
- 121
4
votes
0
answers
93
views
Vershik's conjecture about generic quadratic algebras
Is it still unknown whether very general (lying in a countable intersection of some Zariski opens in corresponding Grassmannian) quadratic algebras $R$ with $\operatorname{dim} R_2 < \frac{3}{4}(\...
- 4,600
4
votes
1
answer
446
views
WZW primary correlations in terms of current algebra?
Given the
$\mathfrak{u}(N)$ algebra
with generators $L^a$ and commutation relations
$ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ ,
the WZW currents of $U(N)_k$
$$ J(z) = \sum_{n \in \mathbb{Z}} J^a_n z^{-n-...
- 545
2
votes
0
answers
137
views
$p$-adic Banach group algebra
Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
- 1,485
0
votes
0
answers
45
views
Reference request for the smallest set( resp. abelian group) that a group ( resp. ring ) has a faithful action on
For a proof of the Cayley's Theorem, it is obvious to see that a group ( resp. ring ) has a faithful action on itself by left-multiplication. I would like to extend the result for a bit and find the ...
- 101
12
votes
0
answers
542
views
Does Wedderburn's Little Theorem hold constructively?
Wedderburn's Little Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative.
The proofs that I am aware ...
- 63.1k
9
votes
1
answer
735
views
Where has this structure been observed?
$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure:
$R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation":
$$R_X (x, y) \cdot R_Y (x +...
- 6,741
2
votes
0
answers
145
views
The "big bracket" in Lie bialgebras
I am looking for a well-written document such as a survey article or textbook that explores the subject of the "big bracket". This concept is briefly introduced in the appendix of Yvette ...
- 442
0
votes
0
answers
99
views
General reference for finite dimensional $*$-algebras over $\mathbb R$?
What references are there for studying finite-dimensional $*$-algebras over the field $\mathbb R$ in their full generality? We assume these are associative and unital.
Note that:
Not every algebra ...
- 4,943
2
votes
0
answers
91
views
Reference request: structure of group of units of finite group ring
Let $G$ be a finite group, let $F$ be a finite field and let $F[G]$ be the group algebra of $G$ over $F$.
What is known about the structure of the group of units $F[G]^\times$? Of course, it must ...
- 1,433
1
vote
0
answers
123
views
Four octonionic loops to identify
A loop is a quasigroup with the identity. I have to disclose that loop-theory is something outside my expertise.
I have four loops arising from octonionic elements of unitary norm that have order 16, ...
- 295
2
votes
0
answers
134
views
Algebra of finite width matrices
$\DeclareMathOperator\FWM{FWM}\DeclareMathOperator\End{End}$For any ring $R$ there's an algebra of finite width matrices with entries in $R$. By finite width matrices I mean the ones that have only ...
- 4,600
3
votes
1
answer
173
views
$\Omega$ for noetherian semiperfect rings
Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-...
- 26.5k
3
votes
0
answers
116
views
A theory of refined h- and f-polynomials for the permutahedra, associahedra, noncrossing partitions, and tropical Grassmannians (references)
Looking for references (insights) on a theory encompassing a notion of refined face polynomials and their associated refined h-polynomials that are generalizations of the relation between ordinary f-...
- 10.5k
3
votes
1
answer
135
views
Does Noetherianity imply division theorem?
I am trying to understand something which is probably basic for experts so I am sorry if this is not suited for this forum.
Let $\mathcal{O}_n$ denote the ring of germs at $0 \in \mathbb{R}^n$ of real-...
- 981
3
votes
1
answer
123
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Equivalent definitions of pro-unipotent coalgebras
I'm trying to find a reference in the literature for equivalence of the following two definitions of pro-unipotent coalgebras.
Definition Let be $H$ a coagumented coalgebra and let $\Delta \colon H \...
