Questions tagged [noncommutative-algebra]
Non-commutative rings and algebras, non-associative algebras. Can be used in combination with ra.rings-and-algebras
522 questions
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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 ...
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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 $A=\...
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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 ...
CommunityBot
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
CommunityBot
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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 $x_ix_j=-x_jx_i$...
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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). ...
CommunityBot
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Ideals in Factors
One can easily prove that factors have no nontrivial ultraweakly closed 2-sided ideals as these are equivalent to nontrivial central projections. One can also show type $I_n$, type $II_1$, and type $...
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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, ...
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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 \...
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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 ...
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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 ...
CommunityBot
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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 ...
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Do torsion-free groups give projectionless group ($C^\ast$) algebras?
One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the ...
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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 ...
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An "Elementary" Math Question Generalized (Ring Theory Perhaps)
The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics"
"Prove that if integers a_1, ..., a_n are all distinct, then the ...
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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 ...
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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 ...
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