Questions tagged [noncommutative-algebra]

Non-commutative rings and algebras, non-associative algebras. Can be used in combination with ra.rings-and-algebras

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1 answer
97 views

Particular example of a quadratic extension of a nonunital ring

I want to construct a concrete non-unital ring $R$ with the following properties: $R$ is a noncommutative non-unital ring with a right unite $r$ i.e $t.r=t$ for any $t\in R$. $S\subset R$ is a ...
1 vote
0 answers
48 views

On a lemma of projective dimension

Let $R$ be a finite-dimensional algebra, and $A=R\oplus A_1\oplus A_2\oplus \dotsb$ be an $\mathbb{N}$-graded algebra which is locally finite (i.e. all $A_i$'s are of finite dimension). Let $\text{...
5 votes
2 answers
386 views

Algebra with three anti-commutator relations

Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra $V$ generated by $ a,b,c$ and the relations $$u a^2 = bc + cb$$ $$v b^2 = ac + ca$$ $$w c^2 = ab + ba$$ Is $V$ generated by ...
9 votes
1 answer
195 views

Matrix ring isomorphisms of different sizes

Do there exist (unital, associative, noncommutative) rings $R$ and $S$, where $\mathbb{M}_2(R)\cong \mathbb{M}_3(S)$, but these matrix rings are not isomorphic to $\mathbb{M}_6(T)$ for any ring $T$?
3 votes
0 answers
122 views

Composition of Frobenius $n$-homomorphisms, characteristic-free?

This question is, as so often, a crossbreed of curiosity and laziness. The former has led me to an interesting, but somewhat unsatisfactory paper by Khudaverdian and Voronov (arXiv:2002.02395v2) and, ...
7 votes
0 answers
189 views

On the structure of an algebra as a bimodule

$\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ker{Ker}\newcommand{\bi}{\mathrm{bi}}\newcommand{\op}{\mathrm{op}}$Let $K$ be a field (say of characteristic zero), and $...
15 votes
1 answer
1k 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) $\mathcal{A}$ and the $C^*$ -algebra of continuous complex-valued ...
1 vote
1 answer
216 views

Wedderburn theorem for finite-dimensional algebras over the complex numbers

I'm trying to understand how to apply the Wedderburn theorem in the context of unitary algebras over $\mathbb{C}$ that are finite-dimensional and semisimple. Let $\mathcal{A}$ be a $\mathbb{C}$-...
7 votes
1 answer
108 views

Semi-simple algebras over operads

I believe people thought about this questions, however I couldn't find any reference. I appreciate if someone could direct me to some detailed discussions about it. The categories of associative ...
9 votes
1 answer
217 views

Formal smoothness of path algebras and connections

Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if $$ \Omega^1_kA = \operatorname{Ker}(\...
2 votes
2 answers
163 views

Minimal ideals and subalgebras of semisimple algebras

I'm considering an algebra to be a ring which is also a vector space over some field $F$, and the algebra $A$ is said to be semisimple if it is semisimple as a ring, i.e., $A$ can be written as a ...
6 votes
2 answers
1k views

Is there a notion of point in noncommutative geometry?

It is not clear to me whether there is a general notion of point in NCG. I have heard (more through physics) that the notion of a point becomes meaningless or ill-defined in noncommutative spaces, but ...
3 votes
0 answers
105 views

On the conditions for Artin-Schelter Gorenstein algebras

Let $ k $ be a field and $ A $ a connected graded $ k $-algebra ($ A $ is associative, but not assumed to be commutative). The algebra $ A $ is called Artin-Schelter Gorenstein* of dimension $ d $ if ...
13 votes
3 answers
1k views

Are the trace relations among matrices generated by cyclic permutations?

Let $X_1,\dots,X_n$ be non commutative variables such that $\operatorname{tr} f(X_1,\dots,X_n) = 0$ whenever the $X_i$ are specialized to square matrices in $M_r(k)$ for any $r \geq 1$. Does this ...
61 votes
3 answers
7k views

What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group. A couple of good ...
1 vote
3 answers
2k 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 ...
5 votes
2 answers
612 views

A note on orders in quaternion algebras

Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$, where $i^2=a,j^2=b;a,b\in F^\times$. ...
8 votes
2 answers
480 views

Faithful flatness and non-commutative algebras

$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following: Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...
6 votes
2 answers
383 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 ...
11 votes
2 answers
1k views

A problem in commutative algebra whose solution requires algebraic geometry (resp., noncommutative algebra)?

One can argue that commutative algebra is affine algebraic geometry. However, a great deal of commutative algebra generalizes to non-commutative algebra, and in that setting there is little geometry, ...
8 votes
0 answers
217 views

Differential birational equivalence

Suppose the base field algebraically closed and of zero characteristic. There are two fascinating questions in the intersection of ring theory and algebraic geometry (for which an excellent discussion ...
3 votes
0 answers
202 views

Arithmetic derivatives and non-commutative generalizations

In the theory of arithmetic derivatives, in the simplest case an arithmetic derivative on $\mathbb{N}$ is defined via the rule $(a \times b)'= a \times b' + a' \times b$, mirroring the product rule ...
5 votes
0 answers
128 views

Confusion around a (necklace) cobracket in Ginzburg's article Calabi-Yau Algebras

Something has been puzzling me for quite a while in Ginzburg's article Calabi-Yau Algebras. At some point he considers the free graded algebra $\mathbb{C}\langle x_1, \dots, x_n, \theta_1, \dots \...
1 vote
3 answers
436 views

Smooth affine algebras are Calabi-Yau

Are all smooth affine algebras over a field Calabi-Yau? I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/
9 votes
0 answers
410 views

Does Wedderburn's Theorem hold constructively?

Wedderburn's 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 of ...
2 votes
1 answer
249 views

Gluing data for modules over a ring with idempotents

Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A\text{-}\mathrm{Mod}$ and $eAe\text{-}\mathrm{Mod}$. This is Example 2.7 in Homological ...
4 votes
0 answers
276 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 ...
1 vote
1 answer
122 views

Polynomial identities satisfied by the Weyl algebra in prime characteristic

The rank $n$ Weyl $A_n(\mathsf{k})$ algebra over a field $\mathsf{k}$ of zero characteristic does not satisfies any polinomial identity. If it were a PI-algebra, Kaplansky theorem would apply (since ...
4 votes
1 answer
232 views

Kaplansky inverse element theorem on group C-star algebra

In a class talking about $C^*$ algebra and (higher) index theory, I heard a theorem (related to Kaplansky, proved?), that is Suppose $\Gamma$ is a group (admitting Haar measure if necessary) while $\...
2 votes
1 answer
237 views

Existence of finite dimensional representation of an algebra

Let $m>1$ be an integer and let $A$ be the algebra generated by the elements $\{u^i_j,v^i_j,\bar{u}^i_j, \bar{v}^i_j| 1\leq i,j\leq m\}$ quotient over the relations \begin{eqnarray} u^i_j v^k_l&...
2 votes
0 answers
131 views

Zero divisors in the extra-special group algebra $\mathbb{R}[2^{1+6}_+]$

Can you characterize the unit-group of the real group-algebra of the extraspecial plus-type 2-group of order 128? (That is $\mathbb{R}[2_+^{1+6}]$ using Conway's notation.) (Please choose any irrep ...
3 votes
0 answers
172 views

The monoid of stably-free modules over integral group rings

Fix a torsion-free group G, let $M_G$ be the monoid of stably-free $\mathbb{Z}G$-modules under operation $\oplus$, the direct sum of modules. In studying objects related to Wall’s D2 problem on CW-...
2 votes
0 answers
67 views

Classification of polynomials leading to finite dimensional admissible algebras

Let $K \langle x , y \rangle $ ($K$ a field, we can assume it has only two elements if it helps) be the non-commutative polynomial ring in 2 variables. Question 1: For which non-commutative ...
24 votes
2 answers
3k 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 ...
1 vote
0 answers
76 views

Inner product on Standard form of von Neumann algebra

Let $(M, H, H_+,J)$ be a standard form of a von Neumann algebra $M$ acting on a complex Hilbert space $H$ endowed with a self-dual cone $H_+$. Is it true that $$\langle x,yz\rangle=\langle zx,y\rangle$...
4 votes
0 answers
67 views

Indecomposable injectives over Weyl algebras

Let $A=A_n(\mathbb{C})$ be the $n$-th Weyl algebra over the complex field. Then $A$ is a left Noetherian noncommutative ring. Is there a complete classification of indecomposable injective $A$-modules?...
27 votes
5 answers
9k views

Can a quotient ring R/J ever be flat over R?

If $R$ is a ring and $J\subset R$ is an ideal, can $R/J$ ever be a flat $R$-module? For algebraic geometers, the question is "can a closed immersion ever be flat?" The answer is yes: take $J=...
2 votes
0 answers
103 views

Correct notion of "connected" for dga of bundle-valued forms

Consider a vector bundle $E$ over a manifold $M$ with flat connection, $\nabla$. From this data I can form the associative/unital differential graded algebra $\mathcal{A} = \left(\Omega^{\bullet}(M, ...
9 votes
1 answer
791 views

Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory

In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
6 votes
0 answers
329 views

Rough paths, unparametrized path space, and Kontsevich's moduli space of stable maps

Let $X$ be a manifold. Modulo reparametrization, the path space of $X$ is a groupoid $\Pi_X$. In Kapranov's "Free Lie Algebroids and the Space of Paths", Kapranov constructs an associated ...
1 vote
1 answer
173 views

A question about surjective maps between quadratic algebras

Let $V$ be a finite-dimensional vector space and $$ U \subseteq W \subseteq V \otimes V $$ be a proper inclusion of vector subspaces. Then take the tensor algebra $$ T(V) = \bigoplus_{i=1}^{\infty} V^{...
1 vote
0 answers
33 views

Number of right divisors of a central skew polynomial

Let $\mathbb{F}$ be a finite field of $p$ elements, $\sigma \in \operatorname{Aut}(F)$ of order $m$, $\mathbb{F}^\sigma$ be the fixed field of $\sigma$, and $\mathbb{F}[x,\sigma]$ be a skew polynomial ...
3 votes
0 answers
140 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 ...
5 votes
0 answers
797 views

Chinese remainder theorem

For non-commutative rings, we have this generalization of the Chinese remainder theorem (CRT). I wonder if there is another statement involving only left or right ideals; do you know any?
0 votes
0 answers
75 views

What is the order of these binary trees?

In the book "Free Lie algebras" by the author Christophe Reutenauer, Example 4.2 (in subsection 4.1) gives the trees of degree $\le 5$ of a Hall set in magma $M(A)$, where $A=\{a,b\}$ as the ...
4 votes
0 answers
96 views

Convolution algebra of a simplicial set

Consider a simplicial set $X^\bullet$ with face maps $d_i$ (assume the set is finite in each degree so there are no measure issues). Then given two functions $f,g:X^1\to \mathbb{C}$ one can form their ...
1 vote
0 answers
58 views

Structure of tame concealed algebra of Euclidean type

I wanted to know some references where people have studied the representation theory of tame concealed algebra of Euclidean type. What do we know about the structure of their module category? What ...
1 vote
0 answers
94 views

Does the center of any finitely generated associative algebra over a field have finite type?

Consider the monoid algebra $R:=K\langle x_1,\dots,x_n\rangle$ generated by $n$ letters $x_1,\dots,x_n$ for $n>1$ over field $K$. Equivalently, $R$ is the tensor algebra $T(V)$ on the $n$-...
4 votes
3 answers
545 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 ...
7 votes
0 answers
563 views

Guises of the noncrossing partitions (NCPs)

From "Noncrossing partitions in surprising locations" by Jon McCammond: Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...

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