Questions tagged [noncommutative-algebra]
Non-commutative rings and algebras, non-associative algebras. Can be used in combination with ra.rings-and-algebras
498
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Is $x \in A_1$ left algebraic over the subalgebra generated by $p$ and $q$, $[q,p]=1$?
Let $A_1:=A_1(x,y,k)$ be the first Weyl algebra over a field $k$ of characteristic zero,
namely, the $k$-algebra generated by $x$ and $y$ with relation $yx-xy=1$.
Let $f:(x,y) \mapsto (p,q)$ be a $k$-...
3
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Language representation problem regarding non-commutative, non-associative algebras
Consider a sentence as a series of words with an associated set of labels that tell one how information is passed through the sentence - examples include combinatory categorical grammars or Lambek ...
6
votes
2
answers
898
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Simple Ore extensions
Let $R[x;\sigma,\delta]$ be an Ore extension, where $R$ is an associative and unital ring and $\sigma : R\to R$ is a (not necessarily injective!) ring endomorphism. (In the literature it is often ...
12
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0
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Hopf-Galois extensions where the "extension" is a module?
For $H$ a Hopf-algebra, an $H$-Hopf-Galois extension is a map of rings $\phi\colon\thinspace A\to B$ such that $H$ coacts on $B$ over $A$, $B\otimes_AB\cong B\otimes H$, and the cofixed points, or the ...
6
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0
answers
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m-systems and n-systems in topological rings
Note that throughout rings have a multiplicative identity and are not necessarily commutative
Definition: Let $R$ be a ring and let $M\subseteq R$. Then, $M$ is an m-system iff for every $x,y\in ...
5
votes
2
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A non-commutative analog of: two polynomials are algebraically dependent iff their Jacobian is zero
Let $f,g \in \mathbb{C}[x,y]$.
There is a well-known result, that can be found for example
here, pages 19-20, that says the following:
$f,g$ are algebraically dependent over $\mathbb{C}$ if and ...
7
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0
answers
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About an algebraic construction of a sheaf of formal microdifferential operators
While reading these notes by Victor Ginzburg on $D$-modules I found a certain construction of Microlocailzation in the algebraic setting which unfortunately doesn't seem to be elaborated on a lot in ...
3
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0
answers
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Intersections of generating sets of subalgebras
Let $A$ be a finitely generated, finitely presented, Noetherian, unital algebra over the complex numbers, which has no zero divisors. We do not assume that $A$ is commutative however.
Moreover, let $...
5
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0
answers
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Which groups can occur as the group of units of finite-dimensional noncommutative algebras?
This is a continuation of a previous question: Connectedness of groups of units in finite-dimensional commutative algebras.
Let $k$ be an algebraically closed field of characteristic $0$. Which ...
6
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1
answer
480
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Non-commutative regular sequences and non-commutative Koszul complex
I'm trying to understand the non-commutative Koszul complex, as can be found in Anick's nice paper "Non-Commutative Graded Algebras and Their Hilbert Series", J. of Algebra 78, (1982) and I'm stuck at ...
2
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0
answers
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group action on Tor groups of modules and smash product
I am trying to understand theorem 3.4.2 from the paper "Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over $\mathbf Z_{(p)}$ for representations with $p$-small weights" by ...
15
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1
answer
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Applications of cluster algebras
Why are so many algebraists nowadays interested in cluster algebras?
(This is a rewording of one half of the closed question Cluster algebras and teichmuller theory.)
4
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1
answer
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If $\{f\in R[x]\:|\:f\text{ monic}\}$ is a right denominator set, is $\{f^i\:|\:i\geq 0\}$ a right denominator set also?
Let $R$ be a right (and left) Noetherian ring and $T=R[x]$ its polynomial ring. It was shown by Stafford that the set $S=\{f\in T\:|\:f\text{ monic}\}$ is a right denominator set. So my question is, ...
12
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Finitely generated skew-fields
There is a well known theorem saying that a commutative field that is finitely generated as a ring has to be finite (Kaplansky).
Is the same true for non-commutative "fields" (usually called ...
1
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0
answers
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Is there a (nontrivial) known example of an algebra over a complete regular local ring with the following property?
I am working on some algebras over complete regular local algebras. But I am not sure whether such rings are worth to study. I am looking for some examples of these algebras. Let $(R,\mathfrak{m})$ be ...
5
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Do almost commutative flat degenerations induce equality in K-theory? (Or: Is the characteristic variety actually a support of a class in $K$-theory?)
I intentionally phrased the title to match a different question which is almost identical to the one i'm asking. However similar, the answer there, which uses commutative algebraic geometry, is not ...
3
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0
answers
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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: ...
4
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3
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Finitely generated projective = finitely presented flat over a noncommutative Noetherian ring
Let $R$ be a possibly noncommutative left Noetherian ring and $M$ an $R$-module. I am looking for a reference or a proof for the following fact: $M$ is finitely generated and projective if and only if ...
6
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0
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Algebraic-closures of division rings
In what follows, $x$ is always taken to commute with the coefficient ring. This means that for any given polynomial, you can put the coefficients to the right or the left of $x$ as you please. This ...
2
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0
answers
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Is there an anti-commutator analog of Zassenhaus formula?
Is anyone familiar with an anti-commutator analog Zassenhaus formula? I have been able to find the anti-commutator analog of the BCH formula
$$e^ABe^A= B + \{B,A\}+\frac{1}{2!}\{\{B,A\},A\}+ \frac{1}{...
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4
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Why should the tensor product of $\mathcal{D}_X$-modules over $\mathcal{O}_X$ be a $\mathcal{D}_X$-module?
Let $R$ be a regular algebra over a field $k$ of char 0. Let $D$ be its corresponding algebra of differential operators.
As in the general setting of non-commutative algebra we can tensor right $D$-...
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Any link between abelian $R/J(R)$ and 2-primal condition
Let $R$ be noncommutative unital ring such that each element of the quotient $R/Soc(R_R)$ is idempotent. If the nilpotent elements of $R$ form an ideal, is it true that the idempotents of $R/J(R)$ ...
2
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1
answer
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If $H$ is essentially equimorphic to $K$, then is $H$ atomic only if so is $K$?
I will first state my question, and then give all the relevant definitions.
Q. Let $H$ and $K$ be monoids, and assume $H$ is essentially equimorphic to $K$. Is it true that $H$ is atomic only if so ...
50
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Invertible matrices over noncommutative rings
Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples?
The question popped up ...
2
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0
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injective dimension of envelope algebras
Let $A$ be a connected graded algebra and $A^o$ the opposite algebra of $A$. Let $A^e=A\otimes A^o$. Suppose that $A$ has a finitely generated projective resolution as a graded $A^e$-module.
My first ...
3
votes
0
answers
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The group of automorphisms and anti-automorphisms of the first Weyl algebra
Let $k$ be a field of characteristic zero, and let $A_1=A_1(k)$ be the first Weyl algebra.
It is well known (first proved by Dixmier, if I am not wrong) that the group of automorphisms of $A_1$, ...
20
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4
answers
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Motivation for the Preprojective Algebra
Let $Q=(Q_0,Q_1)$ be a quiver and $k$ a field. We construct a new quiver $\bar{Q}$ in the following way. Let the vertices of $\bar{Q}$ be the same as the vertices of $Q$, and let the arrows of $\bar{Q}...
5
votes
1
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Examples of noncommutative Bezout domains
I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...
2
votes
1
answer
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Localising a right Noetherian ring at a set of regular elements
Let $R$ be a right Noetherian ring, and $S$ a multiplicative set consisting of regular elements where $1\in S$ and $0\not\in S$. Does the right ring of fractions $RS^{-1}$ exist?
This is what I know ...
0
votes
1
answer
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Right localization of $R[x,x^{-1}]$ at monic $f\in R[x]$
Let $R$ be a right Noetherian ring and $S=\{f\in R[x]\;|\;f\text{ monic}\}$. It is a result of Stafford that $S$ is a right denominator set in $R[x]$, so in particular we can localize $R[x]$ at any $f\...
2
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0
answers
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A certain non-clean ring
I am searching for a non-commutative ring $R$ with identity such that $R$ is not a clean ring and $R/Soc(R_R)$ is a Boolean ring. By a clean ring I mean a ring each of whose elements is a sum of a ...
3
votes
1
answer
95
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Isomorphism concerning $Soc(M_n(R))$
It is known that $M_n(R/J(R))\simeq M_n(R)/M_n(J(R))=M_n(R)/J(M_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R_R)$, where $J(R)$ and $Soc(R_R)$ stand for the Jacobson ...
11
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1
answer
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Determinants of octonionic hermitian matrices
For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying
$a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows.
...
2
votes
0
answers
134
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Is a ring with stable range 2 2-Hermite?
Let $R$ be a (possibly non-commutative) ring. The left stable range of $R$ (denoted $sr_l(R)$) is the smallest $n$ such that every left unimodular row of length $>n$ is reducible. A similar ...
1
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1
answer
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When is $rad(L)[x_1,\ldots]$ radical in $Ker(\varphi_\ast)$?
Suppose we have a local ring $L$ (not necessarily commutative) such that $L/rad(L)$ is a division algebra (here $rad(L)$ is the Jacobson radical of $L$). We clearly have the canonical surjection $\...
2
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1
answer
272
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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 ...
0
votes
0
answers
86
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When does an automorphism extend to a localisation (noncommutative rings)
Let $R$ be a (not necessarily commutative) ring. Let $\tau$ be an automorphism of $R$. Consider the localisation of $R$ at a set of multiplicative elements which satisfy the ore condition, say $X$. ...
4
votes
0
answers
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Is a central simple algebra necessarily cyclic if it splits after a cyclic Galois extension?
Let $A$ be a central simple algebra of degree $n$ over $k$, $\dim_kA=n^2$, let $K/k$ be a cyclic galois extension of degree $n$. Suppose $A\times_kK\cong M_n(K)$, does this imply that $A$ is a cyclic ...
15
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1
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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
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0
answers
217
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What is the motivation behind the definition for a smooth differential graded category?
Let $\mathcal{A}$ be an $\mathbb{F}$-linear differential graded category. It is said to be smooth if it is a perfect complex over the differential graded category $\mathcal{A}^\circ\otimes_\mathbb{F}\...
2
votes
1
answer
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Nonnegatively graded algebra $A$ finitely generated as $k$-algebra iff $A_0$ finitely generated, $A_{>0}$ finitely generated as $A$-module?
This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated ...
3
votes
1
answer
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$M$ is finitely generated as $A$-module iff $M/A_{>0}M$ is finitely generated as $A$ module?
Let $A$ be a nonnegatively graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$.
(Here, given a nonnegatively graded algebra $A$, we've ...
4
votes
1
answer
782
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Left- and right-sided principal ideals of quaternions have same index?
One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the ...
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2
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Explicit description of a quaternion algebra with a prescribed set of ramified places
Let $k$ be an algebraic number field. I understand that given a finite set of non-complex places $S\subset V(k)$ of even cardinality, there exists a unique quaternion algebra $Q$ over $k$ such that $Q$...
0
votes
1
answer
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What are the fixed points of $\alpha^n-\mu_j$ for a fixed $j$?
Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$.
What are ...
113
votes
2
answers
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How would you solve this tantalizing Halmos problem?
$1-ab$ invertible $\implies$ $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 ...
2
votes
1
answer
557
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Why $k[x,y]$ is not a formally smooth algebra?
We could talk about the formal smoothness of an algebra. See for example Ginzburg's lecture notes For an associative algebra $A$ over a field $k$ we define
$$
D(A)=T(A+\bar{A})/(\bar{ab}=a\bar{b}+\bar{...
13
votes
2
answers
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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 ...
7
votes
2
answers
683
views
Quantum Grassmannians?
In noncommutative algebraic geometry a commonly studied family of objects are quantum projective spaces. Theses are certain deformations of the homogeneous coordinate ring of $\mathbb{CP}^n$. For ...
18
votes
1
answer
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Koszul complex for non-Koszul algebras
Let $A$ be a graded, connected, locally finite, quadratic algebra over a field $k$; that is, $A$ may be presented as $T(V)/I$, where $V = A_1$ is a finite dimensional $k$ vector space, and the ideal $...