All Questions
Tagged with ra.rings-and-algebras reference-request
329 questions
4
votes
1
answer
787
views
What is a degenerate Legendre Transformation?
I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...
- 245
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
2
votes
1
answer
203
views
Vertices of a polytope as algebra generators
I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in ...
- 1,488
3
votes
1
answer
343
views
How to define a generating subset for algebra in a category?
As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...
3
votes
1
answer
333
views
What do epimorphisms in noncommutative rings look like?
The question I want to ask is inspired by this mathoverflow post about epimorphisms in the category of commutative rings. I found the seminar (by P. Samuel) referenced by David Rydh particularly ...
5
votes
1
answer
404
views
Equivariant Formality
Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$.
Assume further there exists a $dg$...
- 2,554
4
votes
0
answers
252
views
Formal DG-algebras
Sorry for this question but I really have difficulties with model categories.
Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...
- 2,554
3
votes
0
answers
73
views
Quasi-isomorphisms and Subalgebras
Let $A$ and $B$ $dg$-algebras over $\mathbb{C}$. If there exists an isomorphism $f:A\to B$, then every subalgebra $A'$ of $A$ is isomorphic to the subalgebra $f(A')$ of $B$.
What is if $f$ is only ...
- 2,554
27
votes
6
answers
6k
views
Origin of exact sequences
I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...
- 443
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
2
votes
1
answer
260
views
Universal constructions that factor through endomorphisms
If $\cal A$ is a variety of algebras (e.g., all groups) and $\cal B$ is a subvariety defined by some set of identities $X$ (e.g., abelian groups with $X = \{xy \simeq yx\}$), then there is a functor $...
- 879
4
votes
1
answer
423
views
What is the formula for the commutative multiplication on CP(infinity)?
There is a classic formula for maps $\mathrm{CP}(r) \times \mathrm{CP}(s) \to \mathrm{CP}(r+s)$ or maybe $r+s+1$ using Plücker coordinates - IF memory serves. In the limit we get the abelian ...
- 3,880
7
votes
1
answer
385
views
Morita theorem for simplicial rings
My question is the following: is there an analog of Morita theorem in the simplicial setting?
I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
- 1,276
5
votes
1
answer
481
views
Two definitions of modules in monoidal category
The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here $\...
- 1,276
6
votes
1
answer
413
views
When are infinite dimensional path algebras hereditary?
I allready asked this on MO, but did not get any answer.
Given a finite quiver with relations. When is the path algebra modulo relations hereditary?
If the path algebra is finite dimensional or ...
- 2,554
0
votes
1
answer
96
views
$\mathrm{rk}_R M$ vs $\mathrm{rk}_S M$ - how nice need $R,S$ be?
Let $R\hookrightarrow S$ be Noetherian (noncommutative) rings without zero divisors with $\mathrm{rk}_{R} S < \infty$ (e.g. $S=R*G$ the crossed product of $R$ with a finite group $G$). Let $M$ be a ...
7
votes
1
answer
531
views
Is the definition of Gerstenhaber bracket related to operads?
I was reading these notes by Keller. On the page 19 he defines the Gerstenhaber bracket on the Hochschild cochain complex. But first of all he defines the operation $\bullet$ on the cochains by
$$
\...
- 1,276
6
votes
1
answer
488
views
growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$?
I am interested in the growth rate of this type of group: $G=\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$, where $\sigma(a)=\begin{pmatrix}x&y\\z&w\end{pmatrix}\in SL_2(\mathbb{Z})$, where $a$ is ...
- 1,528
2
votes
1
answer
290
views
Idempotents in Green J classes
I recently read this article Syntactic semigroups. In page $8$, he speaks about a J class having an idempotent is called regular:
A $\mathcal J$-class containing an idempotent is called regular. ...
- 233
1
vote
1
answer
139
views
"Symmetric" Polynomial 4-cocycles
It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)-f(x,z,y)+f(z,x,y)=0$, is always a ...
- 10.4k
3
votes
1
answer
223
views
Thinking about the quadratic dual graphically
Say you have a quadratic algebra $A$ , that is, an algebra defined by a finite list of generators over a ground ring $k$ (either a field or a direct product of fields, which will be assumed $\mathbb{Z}...
- 1,474
11
votes
2
answers
606
views
Temperley-Lieb algebras for other Weyl groups?
The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen ...
- 27.8k
2
votes
1
answer
163
views
Amenable group rings embeddable in skew fields
I've made this question on math.stackexchange.com (also offering a bounty) but I did not receive any answer:
I'm looking for a reference of the following fact:
given a (countable?) amenable group $G$...
- 2,452
3
votes
1
answer
3k
views
Diagonalize the simultaneous matrices and its background [closed]
For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$:
Question1:Is there always a
nonsingular matrix $P$ over the same
field $F$ which ...
- 8,071
1
vote
1
answer
1k
views
Associative algebras with Jacobson radical of codimension 1
Is there a name for finite-dimensional associative $F$-algebras having the Jacobson radical of codimension 1. Of course they are particular local algebras and, indeed, the converse is true provided $F$...
- 99
1
vote
0
answers
102
views
Notion of transversality over the field of Puiseux series.
To a given a Laurent polynomial $f$ over the field of Puiseux seris with parameter $t$, $f \in \mathbb{C} \lbrace\lbrace t \rbrace\rbrace[z_1^{\pm1},...,z_n^{\pm 1}]$, one can associate the ...
- 41
13
votes
3
answers
678
views
IBN for algebraic theories
Let us say that a finitary algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets ...
- 63.1k
2
votes
1
answer
164
views
Algorithmically finite-dimensional (noncommutative) algebras.
Can anyone help to find some information about these structures?
5
votes
2
answers
584
views
BGG-like resolutions and translations
This question originated from my confusion after I read the following paragraph (page 31, section 4.8) in Resolutions and Hilbert series of the unitary highest weight modules of the exceptional ...
- 8,597
4
votes
0
answers
172
views
reference for direct finiteness of the ring of affiliated operators
Let $\Gamma$ be a group, $N(\Gamma)$ its group von Neumann algebra,
$\newcommand{\cUG}{{\mathcal U}(\Gamma)}$
and $\cUG$ the ring of all densely-defined, closed operators $\ell^2(\Gamma)\to\ell^2(\...
- 25.8k
1
vote
1
answer
322
views
translation functors in parabolic category $\mathcal{O}$
I'm looking for a reference for a treatment of translation functors (as defined e.g. in this [question][1]) in parabolic versions of BGG category $\mathcal{O}$.
I am mainly interested in the ...
- 8,597
0
votes
2
answers
288
views
Equivalent Forms of AC
There are many algebraic equivalences of AC in the literature. A famous one is "every ring with identity has a maximal ideal".
Where can I find this equivalences, specially those in rings theory !? ...
user30428
3
votes
0
answers
167
views
A non-commutative ring from SU(2)
$SU(2)$, which will be regarded here as the group of unit quaternions under multiplication, has 3 conjugacy classes of finite subgroups which don't have cyclic subgroups of index 1 or 2. They are:
...
- 3,645
11
votes
4
answers
1k
views
Morita equivalence for *-algebras
This is a reference request. I'm looking for a definition of Morita equivalence of *-algebras, as described below. If anyone thinks that this is not the right way to define Morita equivalence of *-...
- 12.8k
3
votes
0
answers
155
views
Clifford algebra is graded separable
Let $D$ be an algebra of odd differential operators on a free module $V$, this algebra is isomorphic to the Clifford algebra $Cl(V^* \oplus V)$. Let $m$ denote multiplication map $$m : D\otimes D \to ...
- 1,545
13
votes
3
answers
357
views
How should one look at the set of compatible ring structures on a given group?
Earlier today I had a conversation with a friend about ways of putting topologies on sets of first-order structures; we wound up talking about reducts and expansions from a topological point of view, ...
- 21.2k
2
votes
2
answers
301
views
torsion theories localizing the base ring to the same ring
If two torsion theories on a ring localize the ring to the same extension ring, I can find no reason that their "meet" in the lattice of torsion theories must also localize to the same ring. I cannot ...
- 403
5
votes
0
answers
272
views
Cocontinuous monadic functors
Some forgetful functors of algebraic categories preserve colimits, but most do not. In order to understand this phenomen in general, I have classified cocontinuous monadic functors whose target is an ...
- 63.1k
4
votes
1
answer
686
views
Character theory of $2$-Frobenius groups.
This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there.
Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/...
- 1,173
33
votes
2
answers
2k
views
What do cluster algebras tell us about Grassmannians?
One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
- 1,690
3
votes
1
answer
213
views
What is this deformed group algebra named?
I have a semisimple algebra $R$ over a field $k$ that looks like a group algebra $k[G]$ except that it's deformed slightly. That is, instead of a basis $e_1 ..., e_n$ closed under multiplication, I ...
- 148k
4
votes
1
answer
233
views
Generating of the matrix ring by two hermitian matices
Let $p$ be a prime and $q=p^n$. Let $\mathbb F_{q^2}$ be a field with $q^2$ elements and $\sigma$ its authomorhism of order two. A $m$ by $m$ matrix $A$ over $\mathbb F_{q^2}$ is hermitian if $A^\...
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 ...
- 4,000
3
votes
0
answers
1k
views
Does anyone know the definition for Hochschild cohomology of a differential graded algebra?
Let $A$ with $d$ be a differential graded associative algebra ($DG$ algebra).
What is the definition of the Hochschild complex $C^*(A,A)$?
There should be a Hoshschild differential and a cup ...
- 41
6
votes
0
answers
168
views
Classifying algebras with two idempotent generators and involution
Say that a finite-dimensional algebra $H$ over a field $K$ is dihedral if $H$ is generated by idempotents $P_1$ and $P_2$ and there is an algebra involution interchanging $P_1$ and $P_2$.
For example,...
- 1,021
2
votes
1
answer
209
views
Simplicial complex made of central idempotents of an algebra
Let $A$ be an algebra, say over $\mathbb{C}$ and finite-dimensional, but not necessary semisimple. I have the strong feeling, which I would like to prove and use, about the following rather natural ...
- 1,846
11
votes
2
answers
628
views
A quantum Grothendieck group?
Question 1: Given a co-commutative bialgebra, does there exist a sort of Grothendieck group type construction? Presumably this should take the form of a functor from co-commutative bialgebras to hopf ...
- 1,411
11
votes
2
answers
2k
views
Generalizing the Fundamental Theorem of Symmetric Polynomials
The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary ...
- 2,356
1
vote
1
answer
560
views
A good introduction to S unit equations
I was looking up some stuff when I stumbled across S unit equations. It seems to me that they are quite helpful in number theory, as given in this paper.
http://faculty.nps.edu/pstanica/research/...
- 230
0
votes
1
answer
465
views
Weak algebraic structures
The following question can be thought as a sequel of this one.
Here I'm looking for a big list of example of weak algebraic structures: here weak means that the structure (i.e. operations) need not ...
- 3,349