# 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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### Which CAS can do basic non-commutative differential algebra?

This is a repost of my question at MSE from 7 months ago, to which I haven't been able to find an answer yet.
I am looking for a CAS (possibly incl. additional packages/libraries) that can compute ...

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### Trace map on Ext group

Let $R$ be a (possibly non-commutative) unital ring and $M$ be a perfect left $R$-module. Then, we have the trace map
$$
\operatorname{Tr}\colon \mathrm{Hom}_R(M,M)\to R/[R,R]\,.
$$
According to the ...

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### Multiplicative bases, path algebras, and Ext algebras

I am interested in understanding when a multiplicative basis exists for finite dimensional algebras over an algebraically closed field, and, in particular, Ext-algebras that are finite dimensional.
It ...

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### Determining the multiplication via addition and some unary operation

It is known that the addition operation in a skew-field $F$ (more generally, in a quasifield) is uniquely determined by the multiplication operation and the unary involutive operation $1_{-}:F\to F$, ...

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### Is anything known about the center of the Fomin-Kirillov algebra?

Let $\mathcal{B}_{\mathbb{S}_m}$ be the quotient of the Fomin-Kirillov algebra so that its pairing becomes certainly nondegenerate. This algebra is conjecturally isomorphic to the Fomin-Kirillov ...

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### 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 ...

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0
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### 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{...

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### 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 $T$?

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### 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, ...

1
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1
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### 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 $\mathbb{C}$-...

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### 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 ...

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### 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}(\...

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2
answers

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $...

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### 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, ...

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### 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 ...

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### 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 \...

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### 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 $...

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1
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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1
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### 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 $\...

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### 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 ...

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### 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-...

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### 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 ...

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### 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$...

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### 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?...

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### 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, ...

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### 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 ...

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### 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^{...

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### 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 $i^2=a,j^2=b;a,b\in F^\times$.
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $n$-...

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### 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 ...

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### Smallest faithful representation of an upper-triangular matrix quotient

This is a curiosity question that came out of teaching abstract algebra.
Let $F$ be a field, and $n>1$ an integer.
Let $F^{n \leq n}$ be the $F$-algebra of all upper-triangular $n\times n$-matrices ...

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### Finitistic dimension conjecture — why artin algebras?

As I understand it, the finitistic dimension conjecture says that if a ring $A$ is finitely generated over a commutative artinian ring $K$, then the finitistic dimension of $A$ is finite.
My question ...

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### Condition for equality of modules generated by columns of matrices

Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...

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### Projective dimension of group ring

Assume that $G$ is a group and $R$ is a p.i.d. What can we say about the projective dimension of $R[G]$? For example can we say that this dimension is at most $1$ for reductive groups? (I think if $...

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### Relation(s) between units and nilpotent elements in graded noncommutative rings

In Commutative Algebra we have the following standard facts which I am going to state in a slightly different form than usually found in textbooks. Namely, let $A$ be a commutative unital ring of ...

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### Cubical vs. simplicial Hochschild cohomology

Simplicial Hochschild cohomology.
$\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\B}{\mathrm{B}}\newcommand{\Obj}{\mathrm{Obj}}\newcommand{\HH}{\mathrm{HH}}\newcommand{\Mod}{\mathsf{Mod}}$One way to ...

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### Are perfect complexes the same as compact objects in D(R) for noncommutative rings?

The Stacks Project proves Thomason's insight that
compact objects of the derived category $\simeq$ bounded complexes of finitely generated projective modules
in Section 15.78, but the running ...

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votes

1
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### 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&...

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### Regular coherence of tensor algebras

Under what conditions on a bimodule $M$ over a noetherian commutative ring $R$ is the tensor algebra $T=T_R(M)$ regular coherent? A theorem of Gersten says this is true for a free bimodule $M$. If $M$ ...

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### Is there a $3$-commutative algebra?

Let $k$ be a field of characteristic $0$. If $m\ge2$, I denote $P_m$ the standard polynomial in $m$ non-commutating indeterminates:
$$P_m(X_1,\dotsc,X_m)=\sum_{\sigma\in\mathfrak S_m}\epsilon(\sigma)...

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0
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### Abelianization of the group of invertible elements in a finite local ring

Let $R$ be a finite local $\mathbb{F}_q$-algebra. Assume that $R\cong R^*$ as left $R$-modules. Are there any known results about the abelianization $(R^{\times})_{\mathrm{ab}}$?
(We can factor $R$ be ...