The noncommutative-algebra tag has no wiki summary.

**0**

votes

**3**answers

1k views

### How to work with co-multiplication?

Let $C$ be a coalgebra and $\Delta: C \to C\otimes C$ a co-multiplication map. Then, due the co-associative property we can consider $\Delta^m$. But how is defined $\Delta^{m}: C \to C^{\otimes m}$???
...

**-1**

votes

**2**answers

255 views

### Identifying a sequence of polynomials

Studying a specific quantum cluster algebra, I have come across the following sequence of polynomials :
$$X_1$$
$$q^{-1/2}(X_1X_2-1)$$
$$q^{-1/2}(X_1X_2X_3-X_3-X_1)$$
...

**6**

votes

**1**answer

263 views

### Does there exist any massive proper $C^*$-subalgebra?

Definition 1: Suppose $B$ is a $C^* $-algebra. $A$ is massive $C^* $-subalgebra of $B$ iff
1. $A$ is a subalgebra of $B$;
2. for each irreducible representation $\pi$ of $B$ representation $\pi|_A$ is ...

**3**

votes

**0**answers

155 views

### Pulling out factors in a Noetherian Domain

Let $R$ be a Noetherian domain (not-necessarily commutative), and let $S$ be a Noetherian subring of $R$. An element $r\in R$ is left $S$-irreducible if, for any $s\in S$ and $r' \in R$ with $sr'=r$, ...

**4**

votes

**3**answers

661 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**

vote

**1**answer

268 views

### Morphisms of a simple sheaf over an algebra to its double dual

Given a smooth and projective surface $S$ over an algebraically closed field $k$ and a sheaf of Azumaya algebras $R$, i.e. $R$ is a locally free $O_S$-module of finite rank. Let $M$ be a coherent and ...

**4**

votes

**1**answer

775 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 ...

**23**

votes

**1**answer

691 views

### Idempotents in Rings of Differential Operators

Differential Operators on General Commutative Rings
Let k be an algebraically closed field of characteristic zero, and let R be a commutative k-algebra. Then a (Grothendieck) differential operator ...

**6**

votes

**1**answer

279 views

### Simultaneously orthogonally transform two SPD matrices to tridiagonal form?

Supposing you have two SPD matrices $A,B\in\mathbb{R}^{n\times n}$ are there any known results on the existence or non-existence of a unitary matrix $Q$ such that $Q^\top A Q=T_A$ and $Q^\top B Q=T_B$ ...

**1**

vote

**1**answer

252 views

### Q-Divisor and Determinant Map on a Maximal Order

Given a smooth projective surface $X$, let $A$ be a sheaf of maximal orders in a division ring.
Let us for simplicity assume $A$ ramifies in one curve $C$ with ramification index $e$. Let $A^*$ be the ...

**0**

votes

**1**answer

279 views

### Uniqueness of maximal subfields

Let D be a division ring with center Z. Let R and K be two maximal subfields of D, both purely inseparable of exponent one ( means the p power of each of them in Z).
Why are R and K isomorphic?
Or a ...

**8**

votes

**1**answer

522 views

### Mathematical software for computing in integral group rings of discrete groups?

I'm doing computations in the integral group ring of a discrete group,
in particular the discrete Heisenberg group. In this case elements
are integral combinations of monomials $x^k y^m z^n$, where ...

**2**

votes

**2**answers

242 views

### How canonical is the triangular decomposition of a rational Cherednik algebra?

Introduction:
Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, let $G \leq \mathrm{GL}(V)$ be a finite subgroup and let $\kappa:V \times V \rightarrow \mathbb{C}G$ be an alternating bilinear ...

**4**

votes

**0**answers

151 views

### Global dimensions of orders over non-Gorenstein centers

This question concerns the following Lemma 4.2 in this paper by Van den Bergh:
Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that ...

**4**

votes

**2**answers

718 views

### Semiprime (but not prime) ring whose center is a domain

The center of a prime ring is a domain and the center of a semiprime ring is reduced.
Now I have no evidence to believe that if the center of a semiprime ring R is a domain,
then R has to be a ...

**10**

votes

**2**answers

1k views

### Maximal ideal that annihilates entire ring

Does there exist a ring R with a nonzero maximal ideal M such that R^2=R and MR = RM = 0?
Here R is associative but does not have an identity (obviously). It seems a simple enough question but I'm ...

**9**

votes

**2**answers

2k views

### Left and right eigenvalues

A quaternionic matrix $A$ gives rise to a
function $\mathbb{H}^n \to \mathbb{H}^n$
given by $x \mapsto A \cdot x$. This is real linear,
but not complex- or quaternionic-linear
(in general) if we ...

**47**

votes

**2**answers

4k views

### How would you solve this tantalizing Halmos problem?

1-ab invertible => 1-ba invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
Geometric series. In ...

**16**

votes

**3**answers

942 views

### What is the precise relationship between groupoid language and noncommutative algebra language?

I have sitting in front of me two 2-categories. On the left, I have the 2-category GPOID, whose:
objects are groupoids;
1-morphisms are (left-principal?) bibundles;
2-morphisms are bibundle ...

**3**

votes

**2**answers

853 views

### Spectral decomposition for an arbitrary linear combination of position and momentum operators

Suppose we have the Hilbert space L2(Rn) and we have n operators Qi and n operators Pi defined in the usual way by:
Qi ψ(q1,q2,...,qn) = qi ψ(q1,q2,...,qn)
Pi ψ(q1,q2,...,qn) = -i ...

**1**

vote

**1**answer

204 views

### Algebra of endomorphisms of f.g. modules as subquotients of matrix algebras

Let $A$ be a $C$-algebra, where $C$ is a commutative ring with $1$, and $M$ be a finitely generated left $A$-module.
Question: Is it true that we can always find a positive integer $n$, a ...

**7**

votes

**1**answer

337 views

### Semisimple-ish rings!

Let S be the class of all rings R which have 1 and satisfy this condition:
for every "non-zero" right ideal I of R there exists a "proper" right ideal J of R such that I + J = R. (The + here is not ...

**1**

vote

**0**answers

587 views

### What is a “double star-product”

Michel Van den Bergh introduced the notion of a double Poisson algebra. The definition is cooked up such that the representation varieties of such an algebra are Poisson varieties.
Is there a notion ...

**3**

votes

**1**answer

221 views

### Special subalgebras of central simple algebras

In this question F is a field and all algebras are finite dimensional F algebras.
Let X be the set of all F algebras A for which there exist an F algebra B and an F division algebra D such that F is ...

**9**

votes

**1**answer

569 views

### Division algebras in which every proper subfield is maximal

I have a (noncommutative) division algebra D which is finite dimensional over its center F. I know that every subfield of D which contains F properly is a maximal subfield of D. What can we say about ...

**12**

votes

**4**answers

2k views

### What are your favorite finite non-commutative rings?

When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand.
There are many convenient examples of commutative rings, both finite and infinite, ...

**11**

votes

**2**answers

1k views

### when are algebras quiver algebras ?

Good Morning from Belgium,
I'm no stranger to the mantra that quiver-algebras are an extremely powerful tool (see for example the representation theory of finite dimensional algebras). But what is a ...

**13**

votes

**2**answers

802 views

### A Non-Commutative Nullstellensatz

In studying presentations of pro-$p$-groups via generators and relations, one is led (via the so-called Magnus embedding) to questions involving power series in non-commuting variables. Results from ...

**8**

votes

**3**answers

902 views

### construct scheme from quivers?

I heard from some guys working in noncommutative geometry talking about the idea that one can construct the noncommutative space from quivers. I feel it is rather interesting. However, I can not image ...

**11**

votes

**5**answers

2k views

### Definition of an algebra over a noncommutative ring

I've tried in vain to find a definition of an algebra over a noncommutative ring. Does this algebraic structure not exist? In particular, does the following definition from ...

**2**

votes

**1**answer

221 views

### Formal deformations of algebras over not necessarily commutative rings

In Iain Gordon's survery article "Symplectic reflection algebras" the concept of formal deformations of algebras over semisimple artinian (not necessarily commutative) rings is summarized (chapter 2). ...

**3**

votes

**4**answers

712 views

### Homological dimension of a graded ring which is like polynomial ring

Let $k$ be a field of characteristic $0$. Consider the following $k$-algebra $R$, which is the quotient of a tensor algebra generated by elements $x_i$ in degree $1$ with the relation ...

**7**

votes

**1**answer

291 views

### Linear disjointness of subfields of a centrally finite division algbera

I am looking for papers or books which discuss this problem. Thank you for reading:
Let K and L be two subfields of a non-commutative division algebra D with the center Z. Suppose that K and L ...

**12**

votes

**1**answer

455 views

### Tensor products and two-sided faithful flatness

Let $f: R \to S$ be a morphism of Noetherian rings (or more generally $S$ can just be an $R-R$ bimodule with a bimodule morphism $R \to S$). Suppose $f$ is faithfully flat on both sides, so $M \to M ...

**5**

votes

**2**answers

663 views

### Is there a name for this algebraic structure?

I found myself "naturally" dealing with an object of this form:
X is a complex vector space, with a "product" (a,b) → {aba} which is quadratic in the first variable, linear in the second, and ...

**12**

votes

**2**answers

1k views

### What properties “should” spectrum of noncommutative ring have?

There are already a lot of discussion about the motivation for prime spectrum of commutative ring. In my perspective(highly non original), there are following reasons for the importance of prime ...

**8**

votes

**3**answers

584 views

### Existence of non-commutative desingularizations

Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a reflexive $R$-module and let $A=Hom_R(M,M)$. Suppose $A$ has finite global dimension. Then one can view $A$ as a weak ...

**10**

votes

**0**answers

713 views

### Is my definition of a context algebra new?

In my DPhil thesis, I defined what I called a context algebra as a model of meaning in natural language. The idea is to mathematically formalise the notion that meaning is determined by context. It ...

**6**

votes

**2**answers

333 views

### Eigenvalues of an element in a Weyl algebra

I have an operator acting on the polynomial algebra $\mathbb{C}[x,y,z]$ that I would like to find the eigenvalues/eigenvectors of. More specifically, let $P(x_1, \ldots, x_6)$ be a homogeneous ...

**4**

votes

**1**answer

293 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

**1**answer

378 views

### Depth Zero Ideals in the Homogenized Weyl Algebra

Let $\mathcal{D}$ be the $n$th Weyl algebra $ \mathcal{D} :=k[x_1,...,x_n,\partial_1,...,\partial_n] $, where $\partial_ix_i-x_i\partial_i=1$.
Let $\widetilde{\mathcal{D}}$ be its Rees algebra, ...

**5**

votes

**1**answer

243 views

### Classifying Algebra Extensions over a fixed extension?

There are lots of "Ext groups" in homological algebra which measure extensions of various things. I'm sure there must be a homological algebra machine for computing the following, and I'm hoping that ...

**4**

votes

**1**answer

147 views

### When is the essential extension commutes with colimits(or push forward)

Let $M$ be an $R$-module,where $R$ is a hereditary (or cohomological dimension less or equal to 1).Take $E(R)$ to be injective hull of $R$, then we have the essential extension
$i:R^I\rightarrowtail ...

**13**

votes

**1**answer

796 views

### Graded commutativity of cup in Hochschild cohomology

I am trying to get used to Hochschild cohomology of algebras by proving its properties. I am currently trying to show that the cup product is graded-commutative (because I heard this somewhere); ...

**2**

votes

**1**answer

112 views

### Some equivalent statements about primitive algebras

I was reading a paper, and it said that the following were equivalent using the Axiom of Choice, but I tried working it out, and I wasn't sure how: an algebra $A$ is primitive; $A$ has a proper left ...

**13**

votes

**9**answers

1k views

### Examples of noncommutative analogs outside operator algebras?

Theo's question made me wonder if there are other "noncommutative analogs" outside of operator algebras. Some noncommutative analogs from operator algebras include:
A $C^\ast$-algebra is a ...

**8**

votes

**4**answers

645 views

### Morita equivalence and moduli problems

Two rings $A$ and $B$ are said to be Morita equivalent if the category of modules over $A$ and $B$ are equivalent as additive categories. (Here I'm considering left modules).
Ex: $M_n(R)$ (the algebra ...

**6**

votes

**3**answers

238 views

### Inverses in convolution algebras

Let $G$ be a locally compact totally disconnected group, and to make life easy let's suppose its Haar measure is bi-invariant. Let $C_c(G)$ be the space of locally constant complex functions on $G$ ...

**12**

votes

**1**answer

667 views

### Gelfand-Naimark from the category-theoretic point of view

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative C* algebra (with unit) A and the C* algebra of continuous complex-valued functions on its ...

**1**

vote

**3**answers

1k views

### Various Cartan's Lemmata

I am a bit amazed by "Cartan's Lemma".. I have so far seen it in :
Algebraic Geometry sources:
Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...