All Questions
Tagged with noncommutative-algebra reference-request
57 questions
43
votes
8
answers
3k
views
How to quantify noncommutativity?
If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...
- 1,890
21
votes
4
answers
7k
views
Binomial Expansion for non-commutative setting
What could be a reference about binomial expansions for non-commutative elements?
Specifically, where can I find a closed formula for the expansion of $(A+B)^n$ where $[A,B]=C$ and $[C,A]=[C,B]=0$?
...
- 829
21
votes
1
answer
2k
views
Is there any non-commutative ring such that every element other than the identity is a zero divisor?
A (unital) ring $R$ with the property that every element other than the identity $1_R$ is a (two-sided) zero divisor, seems to be commonly called a "$0$-ring" or "$\mathcal O$-ring"...
- 10.5k
13
votes
1
answer
694
views
Classification of long exact sequences
Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.
The category $\mathcal C$ is naturally additive as a subcategory of ...
- 3,184
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
10
votes
3
answers
2k
views
nth term in the Baker-Campbell-Hausdorff formula
I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...
- 245
10
votes
1
answer
1k
views
Explicit isomorphism for quaternion algebras over $\mathbb{Q}$?
It is known that the isomorphism class of a quaternion algebra $A=\binom{a,b}{K}$ over a number field $K$ is determined by the finite set of places $v$ of $K$ where $A\otimes_K K_v$ is a division ...
- 2,851
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
8
votes
1
answer
445
views
For $G=\mathbb{Z}^2\rtimes \mathbb{Z}$, $Spec(\mathbb{Z}G)$=?
Let $G$ be the group $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}=\langle y,z\rangle\rtimes_{\sigma}\langle x\rangle$, where $\sigma(x)=\begin{pmatrix}a, b\\c,d\end{pmatrix}\in SL_2(\mathbb{Z})$, which ...
- 1,528
7
votes
1
answer
281
views
Question concerning the coefficients of block idempotents
Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$.
Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$.
For $\chi\in$ Irr$(G)$ define $e_{\chi} := \frac{\...
- 1,755
7
votes
1
answer
179
views
Symmetry of unique generator property
In this article:
Canfell, M. J. "Completion of Diagrams by Automorphisms and Bass′ First Stable Range Condition." Journal of algebra 176.2 (1995): 480-503.
the author defines a ring $R$ to ...
- 1,507
7
votes
2
answers
362
views
Is there Z_n graded supersymmetry?
I have tried searching for something similar to what is described below, but to no avail. It would be great if somebody could show some right references, where this has been done, or explain why such ...
- 73
6
votes
2
answers
449
views
Survey of recent developments of the Gelfand-Kirillov dimension
It is almost two decades since the now classical books by McConnell and Robinson's
[ Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition. Graduate Studies in ...
- 3,318
6
votes
2
answers
543
views
Rings $R$ such that every [regular] square matrix with entries in $R$ is equivalent to an upper triangular matrix
Let $\text{M}_n(R)$ be the ring of $n$-by-$n$ matrices with entries in a commutative unital ring $R$. Theorem III in
C.R. Yohe, Triangular and Diagonal Forms for Matrices over Commutative ...
- 10.5k
6
votes
1
answer
256
views
Is there any structural characterization of the rings in which every element other than the identity is a (two-sided) zero divisor?
[I fear that I'm missing something obvious here, but I'll dare to ask anyway.]
As we all know, a division ring is a (unital, associative, non-zero) ring where every non-zero element is a unit. So, let ...
- 10.5k
6
votes
1
answer
936
views
Noncommutative HKR theorem
What is the analog of HKR theorem in the noncommutative world?
Recall that the well-known theorem by Hochschild-Kostant-Rosenberg says that for a smooth commutative algebra $A$ of finite type over a ...
- 1,276
6
votes
1
answer
372
views
Cohn localization examples
I'm working on my master's thesis, part of which involves an exposition on Cohn localization. (nlab discussion)
In Free ideal rings and localization in general rings, Sec 7.4, Cohn gives a ...
- 161
6
votes
0
answers
584
views
What are the topics in noncommutative algebraic geometry?
Preface: I know very little about noncommutative algebra and noncommutative geometry, so please feel free to make improvement suggestions for my question. Also, to my knowledge there are several ...
6
votes
0
answers
998
views
Generalized Courant-Fischer theorem
Consider some quaternionic matrix $A$. A right eigvenvalue of $A$ is a quaternion $q$ such that $Ax=xq$ for some $x\in \mathbb{H}^n$. Similarly, a left eigenvalue of $A$ is quaternion $q$ such that $...
- 475
5
votes
2
answers
638
views
Moduli space of modules over non-commutative rings
Let $X=Proj(A)$ be a projective scheme, one can the moduli space of coherent sheaves on $X$ with fixed Hilbert polynomial and stability. Since coherent sheaves on $X$ are all obtained as the ...
- 1,663
5
votes
3
answers
2k
views
Ideal structure of a tensor product of certain algebras
I would be grateful if anyone could give me a reference regarding the following question.
Suppose that $A$ and $B$ are two unital prime algebras over a field $F$ whose center consists of scalar ...
- 423
5
votes
1
answer
2k
views
Length of a module over different rings
Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$.
Under what ...
- 1,391
5
votes
1
answer
911
views
Why Jacobson, but not the left (right) maximals individually?
I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from ...
- 493
5
votes
1
answer
266
views
Rings s.t. each element has a power lying in the center (and their completely prime ideals)
Let $R$ be a ring (throughout, all rings are associative and unital). We say $R$ satisfies condition (C) if, for every $a \in R$, there exists an integer $n \ge 1$ (depending on $a$) such that $a^n$ ...
- 10.5k
5
votes
1
answer
301
views
Discriminants of Clifford algebras
I have a Clifford algebra defined over a field of characteristic not equal to $2$. Is there a formula for its discriminant in terms of the corresponding symmetric bilinear form (or in terms of its ...
- 4,254
5
votes
1
answer
273
views
'Lie correspondence' for formal power series in non-commuting indeterminates
This is related to an earlier question of mine. I would like an argument or a reference (or a missing hypothesis if needed) for the following.
Let $\mathbb{F}\langle\langle \alpha\rangle\rangle$ and ...
- 1,089
5
votes
0
answers
293
views
On the deformation theory of associative algebras
Let us start by recalling the notion of a formal deformation:
Let $K$ be a field of characteristic zero and $A$ be an associative $K$-algebra. Consider a commutative augmented $K$-algebra $R$, with ...
- 541
5
votes
0
answers
442
views
A reference on semisimple linear algebra
Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings?
In fact this question is a ...
- 4,010
4
votes
3
answers
567
views
Gröbner/SAGBI bases for non-commutative setting
It is well known that SAGBI/Gröbner bases are important for commutative and non-commutative algebra. The references for commutative scenery is ample and vast, but I am in trouble to find a good ...
- 829
4
votes
3
answers
1k
views
Set of invertible operators in B(H) is connected. Is it true? Is there a reference?
Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...
- 1,284
4
votes
1
answer
334
views
Non-commutative versions of X/G
Let $X$ be a Riemannian manifold and let $G$ be a (at most countable, if that matters) discrete group acting properly and by isometries on $X$. Let $\mathcal{O}$ be the sheaf of analytic functions on ...
4
votes
2
answers
328
views
What are the main open problems in the theory of quasigroups and loops?
What are the main open problems in the theory of quasigroups and loops?
A short survey would be welcome.
Thanks
4
votes
1
answer
145
views
Example of computation of moduli space of $n$-pointe modules?
I am looking for an example of computation of the isomorphism classes of $n$-point modules over a non-commutative generated graded algebra (assuming all good properties such as Noetherian property). ...
- 41
4
votes
0
answers
289
views
Deformation of modules over noncommutaitve rings
Let $M$ be a finitely generated module over a commutative ring $R$. The first order deformation of module $M$ is parametrized by $Ext^{1}(M,M)$ and the obstruction is parametrized by $Ext^{2}(M,M)$. ...
- 1,663
3
votes
0
answers
151
views
Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion
(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
- 10.5k
3
votes
0
answers
109
views
Noncommutative group schemes corresponding to quantum groups
I'm not an expert on quantum groups by any stretch, so forgive me if this question seems overly naive. That said, I was wondering if there is a way (or if there has been any attempt in the literature) ...
- 1,516
3
votes
0
answers
398
views
Bi-differential operators in the definition of star product in deformation quantisation
Let $X$ be an (affine) Poisson variety (not necessarily smooth) over an algebraically closed field of characteristic 0 (such as $\mathbb{C}$), denote $\mathcal{O}(X)$ its ring of functions and $\{-,-\}...
- 367
3
votes
0
answers
72
views
Reference request: Hecke agebra over non-commutative rings
I think the title sums it up quite well: Is it a useful idea to define the Iwahori-Hecke algebra over a non-commutative $k$-algebra? If so, what shape should the relations attain?
Bonus question: ...
- 281
2
votes
2
answers
105
views
Every non-zero submodule of $R_R$ has an indecomposable direct summand: True when $R$ is von Neumann regular?
Let's say that a (right) module $M$ is well complemented if every non-zero submodule of $M$ has an indecomposable direct summand (by the way, is there a better or more standard name for this property?)...
- 10.5k
2
votes
1
answer
107
views
The ring of upper triangular $n$-by-$n$ matrices over a skew field is (left and right) Rickart
Let $T_n(R)$ be the ring of upper triangular $n$-by-$n$ matrices with entries in a (commutative or non-commutative) unital ring $R$. It happened to me to note that, if $R$ is a skew field, then $T_2(R)...
- 10.5k
2
votes
1
answer
293
views
Notation and reference for polynomials with coefficients not commuting with the indeterminates
Let $R$ be a noncommutative ring (with unit). Then a "fully noncommutative" (for a lack of better wording) monomial over $R$ in the single noncommutative indeterminate $X$ of degree $d$ is given by a ...
- 7,127
2
votes
1
answer
181
views
Relation(s) between units and nilpotent elements in graded noncommutative rings
In Commutative Algebra we have the following standard facts which I am going to state in a slightly different form than usually found in textbooks. Namely, let $A$ be a commutative unital ring of ...
- 7,127
2
votes
1
answer
246
views
The combinatorics of $(f \partial)^n$ in the noncommutative setting?
This is a noncommutative version of these three previous questions:
differential operator power coefficients
Сlosed formula for $(g\partial)^n$
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
...
- 7,127
2
votes
1
answer
287
views
Every module of finite uniform dimension is a direct sum of (finitely many) indecomposable
Crossposted on StackExchange on July 28 (no answer so far).
Let $R$ be a (commutative or non-commutative, associative, unital) ring. It is well known that any artinian or noetherian $R$-module $M$ can ...
- 10.5k
2
votes
1
answer
192
views
Origins of a theorem on an atomic factorizations in domains and cancellative monoids satisfying the ACCPL and the ACCPR
Let $H$ be a (commutative or non-commutative) monoid. We say that $H$ satisfies the ACCPL (ascending chain condition on principal left ideals) if there exists no infinite sequence of principal left ...
- 10.5k
2
votes
1
answer
164
views
Algorithmically finite-dimensional (noncommutative) algebras.
Can anyone help to find some information about these structures?
2
votes
0
answers
98
views
Do $r(a) \leq^\oplus R$ and $r(a) = r(a^2)$ imply $r(a) = eR$ and $aR \subseteq (1-e)R$ for some idempotent $e$?
Let $R$ be a (commutative or non-commutative, associative) ring with unity, and let $a$ be an element of $R$ such that $r(a) = r(a^2)$, where $r(\cdot)$ denotes a right annihilator. It follows that $r(...
- 10.5k
2
votes
0
answers
230
views
Reference request: the UEA of the LR-algebra of tangent vector fields on a smooth manifold coincides with the derivation ring and the ring of diff ops
Let $\mathcal{M}$ be a smooth real manifold and let $A:= \mathcal{C}\left(\mathcal{M}\right)$ be the real algebra of smooth functions on $\mathcal{M}$.
Recall from McConnell, Robson, Noncommutative ...
- 718
2
votes
0
answers
91
views
Fat modules on some algebras.
Let $A$ be a graded $k$-algebra and $M$ a graded right $A$-module. $M$ is called a fat $A$-module if it is generated by degree $0$ and has constant Hilbert polynomial $2$. I wonder for which finitely ...
- 21
2
votes
0
answers
137
views
Noncommutative Castelnuovo-Mumford regularity
I am looking for noncommutative version of Castelnuovo-Mumford regularity. To be more precise, let $A=\oplus_{i=0}^{\infty}A_{i}$ be a $good$ (finite global dimension, connected etc) noncommutative ...
- 21